Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. 2.5 CONTINUITY Intuitively, the graph of a function can be described as a “continuous curve” if it has not breaks or holes. The graph of a function has a break or hole if any of the following conditions occur: The function f is undefined at c The limit of f(x) does not exist as x approaches c The value of the function and the value of the limit at c are different.

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Continuous Function

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Example: Determine whether the following functions are continuous at x=-3. Solution: Observe that f(x) is not continuous at x=-3 since it’s undefined at x=-3, g(x) is not continuous at x=-3 since h(x) is continuous

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Continuity on an interval If a function f is continuous at each number in an open interval (a, b), then we say that f is continuous on (a, b). In the case where f is continuous on, we say f is continuous everywhere. A function is continuous from the left at c if A function is continuous from the right at c if

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved 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 on (a, b) 2.f is continuous from the right at a 3.f is continuous from the left at b. Continuous on a closed interval

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Some properties of continuous functions

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Continuity of polynomials and rational functions Theorem (a)A polynomial is continuous everywhere. (b)A rational function is continuous at every point where the denominator is nonzero, and has discontinuities at the points where the denominator is zero.

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Ex: For what values of x is there a discontinuity in the graph of Solution: The function is a rational function, and hence is continuous at every number where there denominator is nonzero. Solve the equation Yields discontinuities at x=2 and x=3.

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. For example. Find a value of the constant k, if possible, that will make the function continuous everywhere. Solution: since 7x-2 and kx 2 are both polynomials, f is continuous for x 1. x=1 is the only possible discontinuity for f(x). So if k=5, then f is continuous for all x.

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Continuity of compositions For example,

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved Theorem (a)If the function g is continuous at c, and the function f is continuous at g(c), then the composition is continuous at c. (b) If the function g is continuous everywhere and function f is continuous everywhere, then the composition is continuous everywhere. The absolute value of a continuous function is continuous.

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. The intermediate-value theorem

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Approximating roots using the intermediate-value theorem Theorem. If f is continuous on [a, b], and if f(a) and f(b) are nonzero and have opposite signs, then there is at least one solution of the equation f(x)=0 in the interval (a, b).