SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY

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1 (Section 3.2: Derivative Functions and Differentiability) SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY LEARNING OBJECTIVES Know, understand, and apply te Limit Definition of te Derivative Function. Know sort cuts for differentiation, including te Power Rule. Evaluate derivative functions and relate teir values to slopes of tangent lines and rates of cange. Be able to find iger-order derivatives, and recognize notations for various orders. Understand te relationsips between position, velocity, and acceleration in rectilinear motion. Recognize differentiability of functions on open and closed intervals. Recognize possible beaviors of functions and teir graps were tey are not differentiable. PART A: DERIVATIVE FUNCTIONS Let f be a function. f may ave a derivative function, called be given by eiter of te following. f, wose rule may Limit Definition of te Derivative Function (Version 1) f ( x+ ) f ( x) f( x), if it exists 0 Limit Definition of te Derivative Function (Version 2; Two-Sided Approac) f ( x+ ) f ( x ) f( x), if it exists 0 2 We ave taken Limit Definitions from Section 3.1 and replaced te constant a wit te variable x. f ( x+ ) f ( x) x, are variable

2 (Section 3.2: Derivative Functions and Differentiability) Take eiter version. Te domain of f consists of all real values of x for wic te indicated limit exists. We say tat f is differentiable at tose values. is called differentiation. We may say tat f( x) is te derivative of te expression wit respect to x. x is te variable of differentiation. Te process of finding f x derivative of te function f, or tat f x (See Footnote 1.) PART B: SHORT CUTS FOR DIFFERENTIATION We may tink of derivative functions as slope functions. Some Sort Cuts for Differentiation Assumptions: c, m, b, and n are real constants. f is a function tat is differentiable were we care. If g( x)= ten g( x)= Comments f is te 1. c 0 Te derivative of a constant is mx + b m Te derivative of a linear function is te slope. 3. x n nx n1 Power Rule 4. c f x f x c Constant Multiple Rule Proofs. Te Limit Definition of te Derivative can be used to prove tese sort cuts. (See Footnotes 2 and 3.)

3 (Section 3.2: Derivative Functions and Differentiability) Example 1 (Rule 1: Differentiating a Constant [Function]) Let g( x)= 2. Ten, g( x)= 0 (for all real values of x; tis goes witout saying). For instance, g( 1)= 0 and g( )= 0. Observe tat, for eac real value of x, te corresponding point on te grap of g below as a orizontal tangent line, namely te grap itself. Example 2 (Rule 2: Differentiating a Linear Function) Let g( x)= 3x + 1. Ten, g( x)= 3 (for all real values of x). For instance, g( 0)= 3 and g()= 1 3. Observe tat, for eac real value of x, te corresponding point on te grap of g below as a tangent line of slope 3, namely te grap itself.

4 (Section 3.2: Derivative Functions and Differentiability) Example 3 (Motivating Rule 3: Differentiating a Power Function; Evaluating Derivatives and Velocities) Let g( x)= x 3. Unlike in Examples 1 and 2, te derivative function g will not be a constant function. Different tangent lines to te grap of g can ave different slopes. We will use te Limit Definition of te Derivative to find g( x). Tis will parallel our work in Example 2 in Section 3.1. g( x+ ) g( x) g( x) 0 ( x + ) 3 x3 0 0 We will use te Binomial Teorem to expand ( x + ) 3. (See Capter 9 in te Precalculus notes.) ( x) x x x 3 + 3x 2 + 3x x 3 0 3x 2 + 3x () 1 0 ( 3x 2 + 3x + ) 2 1 () x3

5 (Section 3.2: Derivative Functions and Differentiability) x 2 + 3x = 3x 2 + 3x( 0)+ ( 0) 2 = 3x 2 We now ave te derivative function rule g( x)= 3x 2 (for all real values of x). For instance, g()= 1 31 () 2 = 3. We already knew tis from Section 3.1, Part C, but we can now quickly find derivatives for oter values of x. For example, g( 0)= 30 2 = 0, and g( 1.5)= 3( 1.5) 2 = WARNING 1: Do not confuse te original function s values, wic correspond to y-coordinates of points, wit derivative values, wic correspond to slopes of tangent lines. For example, g( 1.5)= 1.5 wic means tat te point 1.5, = g = 3.375, lies on te grap of g. Also, 2 = 6.75, wic is te slope of te tangent line at tat point. In Section 3.1, we saw tat derivatives can be related to rates of cange, suc as velocity. In Section 3.1, Example 7, if te position function is given by st ()= t 3, ten te velocity function is given by v()= t s()= t 3t 2. For instance, v()= 1 3 mp, and v( 1.5)= 6.75 mp.

6 Example Set 4 (Rule 3: Power Rule) (Section 3.2: Derivative Functions and Differentiability) Unless we are instructed to use te Limit Definition of te Derivative, as in Example 3, we will use te Power Rule of Differentiation as a sort cut to differentiate a power of x suc as x 3. We bring down te exponent (3) as a coefficient, and we ten subtract one to get te new exponent, resulting in 3x 2. TIP 1: Be prepared to rewrite a variety of expressions as powers of x so tat te Power Rule may be readily applied. WARNING 2: We will differentiate expressions suc as 3 x and x x in Capter 7. Tey do not represent power functions, and te Power Rule ere does not apply. If g( x)= ten g( x)= x 2 2x 1 = 2x x 3 3x 2 x 4 4x 3 x 17 1 x = x1 1 x = 2 x x x 2 = 1 x 2 2x 3 = 2 x 3 1 x = x 1/2 2 x1/2 = 1 2x, or 1 1/2 2 x 1 x = x 1/3 3 x 2/3 = 1 3x, or 1 2/3 3 3 x 2 1 = 1 4 x = 5/4 x 5/4 5 4 x 9/4 = x 5 In an algebra class, 4 x 9 may be rewritten as x 2 4 ( x ). We will discuss domain issues later in tis section. Te above table demonstrates te following: Te derivative of an even function is odd. Te derivative of an odd function is even. 5 4x, or 5 9/4 4 4 x 9

7 (Section 3.2: Derivative Functions and Differentiability) Example 5 (Rule 4: Constant Multiple Rule) Informally, te Constant Multiple Rule states tat te derivative of a constant multiple equals te constant multiple of te derivative. For example, te derivative of twice x 3 is twice te derivative of x 3. Tat is, if g( x)= 2x 3, ten g( x)= 2( 3x 2 )= 6x 2. TIP 2: Basically, we multiply te coefficient by te exponent, and we ten subtract one from te old exponent to get te new exponent. For instance, g()= 1 61 () 2 = 6.

8 PART C: HIGHER-ORDER DERIVATIVES (Section 3.2: Derivative Functions and Differentiability) Indeed, we can take te derivative of te derivative of, and so on. Higer-Order Derivatives f ( x), read as f double prime of (or at) x, is te second derivative (or te second-order derivative) of f wit respect to x. It is te [first] derivative of f( x) wit respect to x. f ( x), read as f triple prime of (or at) x, is te tird derivative (or te tird-order derivative) of f wit respect to x. It is te [first] derivative of f ( x) wit respect to x. Higer-order derivatives are denoted by f ( 4) ( x), f ( 5) ( x), etc. Roman numerals migt also be used: f IV ( x), f V ( x), etc. (See Footnote 4. See Capter 4 for grapical interpretations of f.) Example 6 (Higer-Order Derivatives) Let f ( x)= x 3. Ten, f( x)= 3x 2, f ( x)= 6x, f ( x)= 6, and f ( 4) ( x)= 0. PART D: RECTILINEAR MOTION: POSITION, VELOCITY, and ACCELERATION In Section 3.1, Parts G and H, we discussed te motion of a car being driven due nort. Tis is an example of rectilinear motion, or motion along a coordinate line. Te starting position of te car corresponds to 0 on te line. Te real numbers on te line correspond to position values, wic are signed distances of te car from te starting position. For instance, 3 corresponds to te position tree miles due nort of te starting point. Positive directions are typically associated wit nort, up, east, rigt, or forward. Also, 3 corresponds to te position tree miles due sout of te starting point. Negative directions are typically associated wit sout, down, west, left, or backward

9 (Section 3.2: Derivative Functions and Differentiability) In our car examples, we let s be te position function for te car. st () gives te position value of te car (in miles) t ours after te trip begins. Te independent variable t represents time or time elapsed. It will be our variable of differentiation. ()= Let v be te velocity function for te car. Ten, vt s (), t because velocity is te rate of cange of position wit respect to time. Te unit of velocity ere is miles per our, or mi, or mp. r ()= ()= Let a be te acceleration function for te car. Ten, at v t s (), t because acceleration is te rate of cange of velocity wit respect to time. Te unit of acceleration ere is miles per our per our, or mi r. 2 Example 7 (Average Acceleration) A commercial says tat a car can go from 0 [mp] to 60 [mp] in 5 seconds. Te average acceleration of te car on tat five-second interval 0, 5 [ ]is given by: v( 5) v = = 12 mp sec We can convert to a more internally consistent unit: 12 mp sec mi / r = 12 sec 3600 sec 1r = 43,200 mi r 2 Example 8 (Position, Velocity, and Acceleration) Let st ()= t 3, as in Example 3. Ten, v()= t s()= t 3t 2, and at ()= v()= t 6t. For instance, a()= 1 61 ()= 6 mi. Tis means tat te 2 r car s acceleration is 6 miles per our per our wen one our as elapsed.

10 PART E: NOTATIONS FOR DERIVATIVES (Section 3.2: Derivative Functions and Differentiability) Let y = f ( x), were f ( x)= x 3, say. Te various orders of derivatives can be denoted in a variety of ways. First derivative Second derivative nt derivative f( x)= 3x 2 f ( x)= 6x f ( n) x (Lagrange s notation) y = 3x 2 (See Warning 3.) dy dx = 3x2 (Leibniz s notation; see note on next page) d dx y = 3x2, or d ( dx x3 )= 3x 2 (See Differential operators note.) D x ( x 3 )= 3x 2 (Euler s notation; see Differential operators note.) (Lagrange s notation) y = 6x (Lagrange s notation) (See Warning 3.) (See Warning 4.) d 2 y dx = 6x d n y 2 dx n (Leibniz s notation; (Leibniz s notation; see Differential see Differential operators note) operators note) d 2 y = 6x, or 2 dx d 2 ( x 3 )= 6x dx 2 (See Differential operators note.) 2 D x ( x 3 )= 6x, or D xx ( x 3 )= 6x (Euler s notation; see Differential operators note.) d n y, or n dx d n ( x 3 ) dx n (See Differential operators note.) n D x ( x 3 ) (Euler s notation; see Differential operators note.) WARNING 3: Te y notation suffers te critical drawback of not indicating te variable of differentiation (ere, x). In tis work, we will assume tat y = dy dx denoted by y. dy. Oter derivatives suc as dt, dy, etc. will not be d WARNING 4: It is not recommended to use y n, since it is easily confused wit y n, te nt power of y. (See Footnote 4.)

11 (Section 3.2: Derivative Functions and Differentiability) Leibniz s notation. Te notation dy evokes te idea of slope. It may be dx tougt of as a quotient of differentials (dy and dx), wic represent infinitesimal (arbitrarily small) canges in y and x. (See Section 3.5.) Separating te differentials is frequently done in practice, altoug many tink of dy as an inseparable entity rater tan a quotient in rigorous work. dx Differential operators. d dx and D x operate on te following expression by differentiating it wit respect to x. Some sources simply use D. Generically, if f is te cubing function, ten Df is tree times te squaring function. d 2 indicates repeated (or iterated ) differentiation. 2 dx For example, d 2 dx y = d d 2 dx dx y. Newton s notation (obsolete). Sir Isaac Newton referred to fluxions, were derivatives were taken wit respect to time: x i = dx dt.

12 (Section 3.2: Derivative Functions and Differentiability) PART F: DIFFERENTIABILITY ON INTERVALS; RIGHT-HAND and LEFT-HAND DERIVATIVES and TANGENT LINES Assume tat f is a function and a and b are real constants suc tat a < b. Differentiability on an Open Interval f is differentiable on te open interval a, b f is differentiable at all real numbers in a, b Tis extends to unbounded open intervals of te form ( a, ), (, b), or (, ). Rigt-Hand Derivative at a Point a; Rigt-Hand Tangent Lines Te rigt-and derivative at a is defined as: f ( a+ ) f a f + ( a) 0 +, if it exists We define te rigt-and tangent line at te point a, f a passing troug tis point wose slope is equal to f + ( a). ( ) to be te line If te above limit can be said to be or, and if f is continuous from te rigt at a, ten te rigt-and tangent line is vertical. Informally, a vertical tangent line indicates were a grap is becoming infinitely steep. (See Footnote 5 on notation.) Left-Hand Derivative at a Point b; Left-Hand Tangent Lines Te left-and derivative at b is defined as: f ( b+ ) f b f ( b) 0, if it exists We define te left-and tangent line at te point b, f b passing troug tis point wose slope is equal to f ( b). ( ) to be te line If te above limit can be said to be or, and if f is continuous from te left at a, ten te left-and tangent line is vertical. (See Section 3.1, Footnote 1 on sign issues.)

13 (Section 3.2: Derivative Functions and Differentiability) Relating One-Sided and Two-Sided Derivatives As wit limits, f( a) exists f + ( a) and f ( a) bot exist, and f + ( a)= f ( a). If c is a real constant, ten f( a)= c ( a)= c and ( a)= c. Differentiability on a Closed Interval f is differentiable on te closed interval [ a, b] 1) f is defined on a, b, 2) f is differentiable on ( a, b), 3) f + ( a) exists, and 4) f ( b) exists 3) and 4) weaken te differentiability requirements at te endpoints, a and b. Imagine taking limits as we pus outwards towards te endpoints. Observe te similarity wit te idea of continuity on a closed interval. Differentiability on alf-open, alf-closed intervals suc as a, b similarly defined. In te case of [ a, b), we would replace a, b in 1), and we would delete 4). f + f [ ) can be [ ] wit [ a, b) [ ] or [ a, b) WARNING 5: Differentiability of f on an interval suc as a, b does not imply differentiability [in a two-sided sense] at a. Tat is, a migt not be in Dom( f ). (Many sources avoid tis issue.)

14 (Section 3.2: Derivative Functions and Differentiability) Example 9 (Differentiability on a Closed Interval) Let f ( x)= 1 x 2 on te restricted x-interval [ 0.8, 0.8]. Ten, f is differentiable on tat interval. Parts of te one-sided tangent lines at te endpoints of te grap of f are drawn in red below. Metods from Section 3.6 will allow us to find f( x). It turns out tat f + ( 0.8)= 4 3 and f ( 0.8 )= 4 3. Example 10 (Vertical Tangent Lines) Let g( x)= 1 x 2 on te implied domain, [ 1, 1]. Ten, g is differentiable on te open interval 1, 1 on te closed interval [ 1, 1]. but is not differentiable Tere is a rigt-and vertical tangent line (in red) at te point ( 1, 0), because te limit of te slopes of te secant lines (in orange) coming in from te rigt can be said to be. Informally, we will write g + ( 1):, g ( 1+ ) g ( 1) because lim =. (It is not, because te secant 0 + lines rise from left to rigt, and we still look at slopes left-to-rigt. ) Also, g is continuous from te rigt at 1.

15 (Section 3.2: Derivative Functions and Differentiability) Tere is a left-and vertical tangent line (in red) at te point ( 1, 0), because te limit of te slopes of te secant lines (in orange) coming in from te left can be said to be. Informally, we will write g (): 1, because lim 0 at 1. g ( 1+ ) g 1 () PART G: NON-DIFFERENTIABILITY =. Also, g is continuous from te left We will examine a variety of situations in wic a function f is not differentiable at a real constant a. Tat is, f a does not exist (DNE). Differentiability Implies (and terefore Requires) Continuity 1) If f is differentiable at a, ten f is continuous at a. 2) If f is not continuous at a, ten f is not differentiable at a. 1) and 2) form a pair of contrapositive statements. Terefore, tey are logically equivalent. Since 1) is true, 2) must also old true. Footnote 6 as a proof. Example 11 (Differentiability Requires Continuity) Let f ( x)= 1, x > 1. Ten, f is not continuous at x = 1. 1, x 1 Terefore, f is not differentiable at x = 1. Te slopes of te secant lines coming in from te rigt at x = 1 f ( 1+ ) f () 1 approac, so lim =, and f () 1 does not exist 0 + (DNE). However, because f is not continuous from te rigt at x = 1, its grap does not ave a rigt-and vertical tangent line at te point ( 1, 1).

16 (Section 3.2: Derivative Functions and Differentiability) We now consider situations were a function is continuous at a, but it is not differentiable tere. Tis typically means tat its grap makes a sarp turn at x = a (indicating two tangent lines) or it as a vertical tangent line tere. Losing Differentiability at a Corner Assume tat f is continuous at a. f is not differentiable at a, and its grap as a corner at te point ( a, f ( a) ) 1), 2), or 3) below olds: 1) f + ( a) and f ( a) bot exist, but f + ( a) f ( a), 2) f + ( a) exists and f ( a): ± (left-and tangent line is vertical), or 3) f + ( a): ± (rigt-and tangent line is vertical) and f ( a) exists A point on a grap is a corner tere are two distinct tangent lines tere, one from eac side. Example 12 (Losing Differentiability at a Corner; Derivatives of Piecewise-Defined Functions) Let f ( x)= x = x, x 0. x, x < 0 on We can use our basic rules to differentiate te different rules for f x teir different subdomains (indicated by x 0 and x < 0 ), altoug we must investigate values of x were te rule for f ( x) canges (ere, at x = 0 ). f( x)= 1, x > 0 1, x < 0 Altoug f is continuous at x = 0, we can see from te grap of f below tat f is not differentiable tere, and tere is a corner at te origin. Tis is because f + ( 0)= 1, wile f ( 0)= 1. Grap of f Grap of f

17 (Section 3.2: Derivative Functions and Differentiability) Here is te proof tat f + ( 0) 0 + Here is te proof tat f ( 0) 0 f + ( 0)= 1: f ( 0 + ) f ( 0) 0 + f ( 0)= 1: f ( 0 + ) f ( 0) 0 f 0 f = 1 0 = 1 Example 13 (Losing Differentiability at a Corner wit a Vertical Tangent Line) Let f ( x)= x2, x < 0 x, x 0. Ten, f 2x, x < 0 ( x)= 1 2 x, x > 0. Altoug f is continuous at x = 0, we can see from te grap of f below tat f is not differentiable tere, and tere is a corner at te origin. Tis is because f + ( 0):, wile f ( 0)= 0. Grap of f Grap of f Here is te proof tat f + ( 0) f + ( 0): : f ( 0 + ) f ( 0) 0 + Limit Form: = f 0 0 +

18 (Section 3.2: Derivative Functions and Differentiability) Here is te proof tat f ( 0) 0 0 = 0 f ( 0)= 0 : f ( 0 + ) f ( 0) 0 f An informal sort cut. If f is a simple function tat is continuous at a, a one-sided derivative (even informally as ± ) can be guessed at by taking te corresponding one-sided limit of te derivative as x a. Tat is, we tentatively guess tat f + ( a) f( x), and f x a + ( a) f( x), x a were and are possible informal results. Instead of taking te limit of slopes of secant lines, we are taking te limit of slopes of tangent lines. Here: Guess: lim f( x) x x wic suggests tat f + ( 0):, and x Limit Form: =, Guess: lim f x 0 wic suggests tat ( x) x 0 f ( 0)= 0. ( 2x )= 0, However, tis trick does not work for more complicated functions! (See Footnote 7.)

19 (Section 3.2: Derivative Functions and Differentiability) Losing Differentiability at a Cusp Assume tat f is continuous at a. f is not differentiable at a, and its grap as a cusp at te point ( a, f ( a) ) 1) or 2) below olds: 1) f + ( a): and f ( a):, or 2) f + ( a): and f ( a): Te rigt-and and left-and tangent lines at a cusp are bot vertical, but te secant lines coming in from one side fall, wile te secant lines coming in from te oter side rise. Example 14 (Losing Differentiability at a Cusp) Let f ( x)= x 2/3, or 3 x 2. For graping purposes, observe tat f is an even, nonnegative function wit domain. Altoug f is continuous at x = 0, we can see from te grap of f below tat f is not differentiable tere, and tere is a cusp at te origin. Tis is because f + ( 0):, wile f ( 0):. Grap of f Wit secant lines (in orange) and tangent line (in red) Here is te proof tat f + ( 0) f + ( 0): : f ( 0 + ) f ( 0) 0 + 1/ f 0 2/3 0 + Limit Form: =

20 Here is te proof tat (Section 3.2: Derivative Functions and Differentiability) f ( 0) f ( 0): : f ( 0 + ) f 0 1/ Using te informal sort cut. Guess: lim x 0 + wic suggests tat Guess: lim x 0 wic suggests tat f( x) 0 f 0 Limit Form: 1 0 f( x)= 2 3 x1/3 = 2 x x f + ( 0):, and f( x) 2 x x f ( 0): x. 2/3 0 = Limit Form: Limit Form: 2 0 =, =, Example 15 (Losing Differentiability at a Point wit a Vertical Tangent Line) Let f ( x)= x 1/3 3, or x. For graping purposes, observe tat f is an odd function wit domain. Altoug f is continuous at x = 0, we can see from te grap of f below tat f is not differentiable tere, and tere is a vertical tangent line at te origin, even toug tere is neiter a corner nor a cusp tere. Tis is because f + ( 0): and f ( 0):. Grap of f Wit secant lines (in orange) and tangent line (in red)

21 (Section 3.2: Derivative Functions and Differentiability) FOOTNOTES 1. Functions tat are nowere differentiable. Functions tat are nowere continuous are also nowere differentiable. See Footnote 3 in Section Proof of te Power Rule of Differentiation. An elegant proof of te Power Rule for all real constants n will be found in te Footnotes for Section 7.5; it will employ Logaritmic Differentiation. Some sources use te Binomial Teorem to first prove it for positive integers n. n = 0 corresponds to te special case of differentiating 1; 0 0 is often defined to be 1 for tis purpose. For positive rational values of n, let n = p, were p and q are positive q integers. Let y = x n. Ten, y = x p/q, and y q = x p. Te Implicit Differentiation tecnique from Section 3.7 can be used to prove te Power Rule in tis case. For negative rational values of n, let m = n. Ten, x n = x m = 1, and te Reciprocal (or Quotient) Rule of m x Differentiation in Section 3.3 can be applied. In tese last two cases, te domain of te derivative migt not be. 3. Proof of te Constant Multiple Rule of Differentiation. Assume tat f is a function tat is differentiable were we care, and c is a real constant. Let g( x)= c f( x). g( x+ ) g( x) c f( x+ ) c f( x) c g( x) ( We will exploit te Constant Multiple Rule of Limits. ) f( x+ ) f( x) = c lim = c f 0 ( x) f( x+ ) f x 4. f n notation. We use f ( n) to denote an nt-order derivative, as opposed to f n. Tis is because n often represents an exponent in te notation f n, except wen n = 1 (in wic case we ave a function inverse). For example, f 2 is often taken to mean ff ; tat is, f 2 ( x)= f ( x) f ( x). For example, we will accept tat sin2 x = ( sin x) ( sin x), wic is te standard interpretation. Note: f x f x f x. is typically not equivalent to On te oter and (and tis compounds te confusion), some sources use n to indicate te number of applications of f in compositions of f wit itself; te result is called an iterated function. For example, tey would let f 2 = f f, and tey would use te rule: f 2 ( x)= f ( f ( x) ). Tis is typically different from te rule f 2 ( x)= f ( x) f ( x). However, our use of te notation f 1 for f inverse is more consistent wit tis second interpretation, since f 1 f is an identity function, wic could be construed as f 0 in tis context. Note: f( x) is typically not equivalent to f ( f( x) ).

22 (Section 3.2: Derivative Functions and Differentiability) Notation for rigt-and and left-and derivatives. Tere appears to be no standard notation for rigt-and and left-and derivatives. In fact, f + ( a) sometimes denotes te upper rigt Dini derivative at a, wic is a bit different from wat we are calling a rigt-and derivative. If te upper rigt Dini derivative at a and te lower rigt Dini derivative at a exist and are equal, ten teir common value is te rigt-and (or rigt-and Dini) derivative at a. Likewise, if te upper left and lower left Dini derivatives at a are equal, ten teir common value is te left-and derivative at a. See T.P. Lukasenko, Dini derivative, SpringerLink, Encyclopedia of Matematics, Web, 4 July 2011, <ttp://eom.springer.de/>. 6. Proving tat differentiability implies (and requires) continuity. f is differentiable at a f( x) f( a) lim exists; see Version 1 of te Limit Definition of f xa ( a)in Section 3.1. x a A rigorous approac: Assume tat f is differentiable at a. Tis implies tat f( a) exists; in fact, it implies tat f is defined on an open interval containing a. lim f( x) f( a)+ f( a) f( x) f( a) ( x a) f x xa xa xa x a + f a f( x) f( a) = lim xa x a lim x a xa + f ( a )= f ( a) 0 + f( a)= f( a). Terefore, lim f x xa = f( a), wic defines continuity of f at a. An intuitive approac: As x a, x a 0. Te only way te limit can exist as a real number is if it as te Limit Form 0 DNE. (Te Limit Form cannot yield a real number c 0 0 as a limit, basically because c 0 = 0.) Tis requires tat f( x) f( a) 0 as x a. Tis, in turn, requires tat f( x) f( a) as x a, or lim f x xa = f( a), wic defines continuity of f at a. 7. A function wit a derivative tat is defined but discontinuous at 0; failure of te limit of te derivative sort cut for one-sided derivatives. See Gelbaum and Olmsted, Counterexamples in Analysis (Dover), p.36. Let f( x)= x2 sin 1 x, x 0. 0, x = 0 Ten, f( x)= 2xsin 1 x cos 1 x, x 0. 0, x = 0 Te metods of Sections can be used to work out te top rule. Wy is f ( 0)= 0? f ( 0 + ) f ( 0) 2 sin 1 0 f ( 0) sin = 0 by te Squeeze (Sandwic) Teorem from Section 2.6. f is discontinuous at 0, because lim f x x0

23 (Section 3.2: Derivative Functions and Differentiability) does not exist (DNE). Also, because lim x0 + f ( x) does not exist (DNE) and lim x0 f ( x) does not exist (DNE), te limit of te derivative sort cut for guessing one-sided derivatives (even informally as ± ), as described in Example 13, fails for tis example. (See also Section 3.6, Footnote 4.) Grap of f Grap of f (Axes are scaled differently.)