Continuity Intuitively, a function is continuous if its graph can be traced on paper in one motion without lifting the pencil from the paper. Thus the graph has no tears or holes. To handle complicated functions, particularly those for which we have a reasonable formula or formulas, we need a more precise definition. Definition. A function f is continuous at a point a if 1. f(a) is defined, that is a is in the domain of f. 2. The function f has a limit as x approaches a. 3. lim fx () = fa () x a
a Limit exists at a, but value does not.
a Here f(a) exists, but the limit does not.
Here f(a) exists, but the limit does not. a
a Here the limit exists at a, and the value also exists, but they are not equal.
Definition: f is continuous on an interval (a, b) if it is continuous at all points of that interval. Example 1. (a) Polynomials are continuous everywhere (b) Rational functions are continuous at all points where the denominator is not 0. Example 2. f(x) = x is continuous everywhere. The piecewise formula definition on f is as follows: xx 0 f() x = xx < 0 Thus if a is positive, then in some interval around a, f(x) is identical with x, and so lim f() x = lim x = a = f(a). x a x a So f is continuous at a.
If a is negative, then in some interval around a, f(x) is identical with x, and so lim f() x = lim x = a = f(a). x a x a So f is continuous at a. Finally, at 0, we have (for reasons similar to those mentioned above) lim f() x = lim x 0 + x 0 + x = 0, and lim f() x = lim x x 0 x 0 = 0, therefore lim () 0 x 0 f x = Since f(x) = 0, the function is also continuous at 0, and so is continuous everywhere.
It is clear that the graph is not broken, nor does it have holes.
The following theorem follows immediately from the main theorem on limits. Theorem. Suppose that functions f and g are continuous at c. Then (a) f + g is continuous at c, (b) f g is continuous at c, (c) f g is continuous at c, (d) f / g is continuous at c, if g(c) 0, and it has a discontinuity at c if g(c) = 0. We show how to prove part (c), and leave the others. Part (d) is proved in the text. lim f()() xgx = lim f() x lim gx () x a x a x a = f()() aga
Composition g(x) f(x) ( f g)( x) = f(()) gx Example: 2 gx () = x+ 1 f() x = 2x f gx () = f( gx ( ) = 2( x+ 1) 2= 2x+ 2 g f() x = g( f()) x = f()1 x + = 2x 2 + 1
Example. Let f(x) = sin(x), g(x) = x 2. Compute: f g, g f, f f, g g. ( f g) x = f gx = gx = x2 ( ) = = 2= 2= 2 g f () x g( f()) x f() x (sin( x)) sin (). x ( ) () (()) sin( ( )) sin( ) f f () x = f ( f ()) x = sin( f ()) x = sin(sin( x )) ( g g )() x = ggx ( ( )) = gx () 2= ( x 22 ) = x 4
Continuity of Compositions The following result is useful for calculating limits of composite functions. Theorem. Let lim stand for any of the following: lim lim x a x a + lim lim lim x a x + x Then if lim g(x) = L, and if f is continuous at L, we have lim f(g(x)) = f(l).
Example. Let f(x) = x. We know that f is continuous everywhere. It follows that if lim g(x) = L, then lim g(x) = L. Thus lim 4 + x3 = lim (4 + x3) x 2 x 2 = (4 + ( 2) 3) = 4 = 4 Also lim sin( x) + 1 = 1 = 1 x 0
Example. Find any points of discontinuity for the functions below. 1. f() x = 2 3x + x 5x 2 3 6 Continuous everywhere, since it is a polynomial. 2. f() x = x+ 3 = x+ 3 2 x + 3x ( x+ 3) x Thus f(x) = 1 x> 3 x 1 x< 3 x Possible problems at 0 and at 3. At 0 it tends to infinity from both sides, so it does not have a finite limit, or a value.
At 3 it has different limits from the right and left and no value. Thus the points of discontinuity are at 0 and 3.
Continuity from the left and right. If we only have right hand or left hand limits, we can define a similarly one-sided version of continuity. Definition: (a) We say that a function f is continuous from the right at a number c if lim f () x = fc x c () + (b) We say that a function f is continuous from the left at a number c if lim f() x = fc () x c Note that f is continuous at c if and only if it is continuous from the right at c and from the left at c.
c Here f is continuous from the right, but not from the left.
c Here f is continuous from the left, but not from the right.
Here f is not continuous from the left or the right. c
Definition. A function f is said to be continuous on a closed interval [a, b] if the following conditions are satisfied: 1. f is continuous at each point of (a, b). 2. f is continuous from the right at a. 3. f is continuous from the left at b. The next slide shows a typical picture of a function defined and continuous on a closed interval.
f(x) a b
Example. 2 f() x = 4 x The natural domain of this function is the closed interval [ 2, 2]. For any point c in the open interval ( 2, 2) (that is 2<c<2) we have: 2 lim f() x = lim 4 x = lim 4 x2 = 4 c 2 = fc () x c x c x c At the left end point 2, we have 2 lim () lim 4 x = lim 4 x 2 f x = + x x 2 + x 2 + 2 = 0 = f ( 2) A similar computation holds at the right endpoint 2.
This is the graph.
The Importance of Continuity in a Closed Interval The intermediate Value Theorem. If f is continuous in a closed interval [a, b], and k is any number between f(a) and f(b), then there is at least one number x in the interval [a, b], so that f(x) = k. f(a) k f(b) a b
f(a) k f(b) a b Possible choices for the number x in the intermediate value theorem.
Corollary: If f is continuous in a closed interval and its values at the end points take opposite signs, then f(x) = 0 for some x in the interval. Example. Every polynomial of odd degree has at least one real zero.
Of course the intermediate value theorem is not necessarily true without continuity. k a b