Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 1 - Limits and Their Properties - 1.4 Exercises - Page 80: 58

Answer

The function is continuous everywhere except at the nonremovable discontinuities$$x= \{ 2n+1 \mid n \in \mathbb{Z} \}.$$

Work Step by Step

As we know, the trigonometric functions $\sin x$ and $\cos x$ are continuous everywhere, so the function $f(x)= \tan \frac{ \pi x}{2}=\frac{ \sin \frac{ \pi x}{2}}{\cos \frac{ \pi x}{2}}$ is continuous everywhere except at those points vanishing the denominator. So the discontinuities of the function are$$\cos \frac{\pi x}{2}=0 \quad \Rightarrow \quad x= \{ 2n+1 \mid n \in \mathbb{Z} \}$$These discontinuities are nonremovable since as we approach these points, $\tan \frac{ \pi x}{2}$ tends to infinity, so we cannot redefine the function at these points.
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