Continuity MATH 1003 Calculus and Linear Algebra (Lecture 13.5) Maoseng Xiong Department of Matematics, HKUST A function f : (a, b) R is continuous at a point c (a, b) if 1. x c f (x) exists, 2. f (c) exists, 3. x c f (x) = f (c). f (x) is called continuous over (a, b) if f is continuous at every point c (a, b). 0 Te function f (x) is discontinuous at x = 1, 2 but is continuous at all oter points: f (x) is not defined at x = 1, and x 2 f (x) does not exist. 1 Is te function f (x) continuous at te point? 1. f (x) = x 1 at x = 2. x 2 1 if x > 2 2. f (x) = 3 if x = 2 x + 1 if x < 2, at x = 2. Solution 1. Since x 2 (x 1) = 2 1 = 1 and f (2) = 2 1 = 1, tey are equal, so f is continuous at x = 2. 2. x 2 f (x) = x 2 (x + 1) = 2 + 1 = 3, x 2 + f (x) = x 2 +(x 2 1) = 2 2 1 = 3, f (2) = 3, tey are equal, so f is continuous at x = 2. Remark For question 2, if f (2) is canged to 5, ten f is not continuous at x = 2.
Continuity properties Continuous functions Teorem 1. A polynomial function is continuous for all x (f (x) = 3x + 5 + 8x 10 is continuous for all x). 2. A rational function is continuous for all x except tose values tat make a denominator 0. (f (x) = x2 +5 x 1 is continuous for all x 1) 3. If n is an odd positive integer, ten n f (x) is continuous werever f (x) is continuous. ( 3 x 4 is continuous for all x) 4. If n is an even integer, ten n f (x) is continuous werever f (x) is continuous and nonnegative. ( 4 x is continuous on [0, )) Teorem If f (x) is continuous on (a, b) and f (x) 0 for all x in (a, b), ten eiter f (x) > 0 for all x in (a, b) or f (x) < 0 for all x in (a, b). In oter words, if f (x 1 ) < 0 and f (x 2 ) > 0 for a continuous function f, ten tere exists x 0 suc tat f (x 0 ) = 0. Solving inequality and sign carts Let y = f (x). Rougly speaking, te rate of cange of y is Find te range of x suc tat: x+1 1. x 2 x 2. 2 1 x 3 x 3. 2 +1 x 3 > 0 (Answer (, 1) (2, )) < 0 (Answer (, 1) (1, 3)) > 0 (Answer (3, )) Cange in y Cange in x More rigorously, we ave te following definition: For y = f (x), te average rate of cange from x = a to x = b is f (b) f (a) b a
A small ball dropped from a tower will fall a distance of y feet in x seconds, as given by te formula y = 16x 2. (a) Find te average velocity from x = 2 seconds to x = 3 seconds. (b) Find te average velocity from x = 2 seconds to x = 2 + seconds, 0. (c) Find te expression from part (2) as 0, if it exists. Solution (a) Te average velocity is 16 32 16 2 2 second. 3 2 (b) Te average velocity is 16(2 + )2 16 2 2 2 + 2 feet per second. (c) 16( 2 + 4) = 80 feet per = 16(2 + 4) = 16( + 4) = 64 feet per second. Instantaneous Rate of Cange In te previous example, we consider te average rate of cange of distance wen te cange of x is from 2 to 2 +. And ten we let tends to 0. Te it can be regarded as te instantaneous rate of cange of at 2. In general, we ave te following definition: For y = f (x), te instantaneous rate of cange at x = a is f (a + ) f (a) i.e. it is te it of te difference quotient of f at x = a. Te instantaneous rate of cange at x = 2 in te previous example is 64. A line troug two point on te grap of y = f (x) is called a secant line. If (a, f (a) and (a +, f (a + )) are two points on te grap of y = f (x), ten te slope of secant line from x = a to x = a + is f (a + ) f (a) (a + ) a = f (a + ) f (a). Tus, te slope of secant line can be interpreted as te average rate of cange of y from x = a to x = a +.
Given y = f (x) = 0.5x 2, (a) Find te slope of secant line for a = 1, and = 2. (b) Find te slope of secant line for a = 1 and for any nonzero number. (c) Find te it of expression in (b) as 0. Solution (a) Te slope of secant line is (b) Te slope of secant line is f (1 + ) f (1) (c) As 0, we ave f (3) f (1) 2 = 0.5(1 + )2 0.5 = 2 = + 0.52 = 1 + 0.5 f (1 + ) f (1) = (1 + 0.5) = 1 Slope of a Tangent Te Derivative From te grap, we observe tat te slope of te secant line tends to te slope of te tangent as tends to 0. Terefore, we ave te following definition: Given y = f (x), te slope of te tangent line of f(x) at te point x = a is given by if te it exists. f (a + ) f (a) For y = f (x), we define te derivative of f at x, denoted by f (x), dy dx df or, to be dx f (x) = f (x + ) f (x) if te it exist. If f (x) exists for eac x in te interval a < x < b, ten f is said to be differentiable over a < x < b.
Summary Exercise Tere are tree different interpretations of te derivative of f (x): Limit of te difference quotient: f (x) is te it of te different quotient of f at x. Slope of te tangent line: f (x) is te slope of te line tangent to te grap of f at te point (x, f (x)). Instantaneous rate of cange: f (x) is te instantaneous rate of cange of y = f (x) wit respect to x. Find f (1) for eac of te following functions: (a) f (x) = 2x x 2 (b) f (x) = x 3 (c) f (x) = 1 x (d) f (x) = x Answers: a) 0 b) 3 c) -1 d) 1/2