The function f(x)=cases4+^-1x, & |x|112(|x|-1), & |x|1cases is:

Question

The function $f(x)=\begin{cases}\dfrac\pi4+	an^{-1}x, & |x|\le1\\dfrac12(|x|-1), & |x|>1\end{cases}$ is:

Accepted Answer

At $x=1$: LHL $=\dfrac\pi4+\dfrac\pi4=\dfrac\pi2$, RHL $=0$ --- discontinuous. At $x=-1$: LHL $=0$, RHL/value $=\dfrac\pi4-\dfrac\pi4=0$ --- continuous but the slopes differ (corner), so non-differentiable at $x=-1$. So $f$ is continuous on $\mathbb{R}-\{1\}$ and differentiable on $\mathbb{R}-\{-1,1\}$.

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