Question

Let $f\left(x\right)=\{\begin{array}{c}\cos x,ifx\ge 0\\ -\cos x,ifx<0\end{array}$
which one of the following statements is not true?

• A. $f\left(x\right)$ is continuous at $x=1$
• B. $f\left(x\right)$ is continuous at $x=-1$
• C. $f\left(x\right)$ is continuous at $x=2$
• D. $f\left(x\right)$ is continuous at $x=-2$
• E. $f\left(x\right)$ is continuous at $x=0$

Answer 1

Answer: (E)

$f\left(x\right)$ is continuous at $x=0$

Given, $\underset{x\to 0^{+}}{lim}xf\left(x\right)=\underset{h\to 0}{lim}\cos h=1$
and $\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}-\left[\cos (-h)\right]$

$=\underset{h\to 0}{lim}-\left(cosh\right)=-1$
So, $f\left(x\right)$ is continuous at $x=0$