At \(x=1\), the function \(f(x)=\left\{\begin{array}{cc}x^{3}-1, & 1 < x < \infty \\ x-1, & -\infty < x \leq 1\end{array}\right.\) is

\(f(x)=\left\{\begin{array}{cc}x^{3}-1, & 1 < x < \infty \\ x-1, & -\infty < x \leq 1\end{array}\right.\)
We have to check the continuity at \(x=1\). \(\mathrm{RHL} \Rightarrow \lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1^{+}}\left(x^{3}-1\right)=1-1=0\) \(\mathrm{LHL} \Rightarrow \lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{-}}(x-1)=1-1=0\) \(f(\mathrm{l})=1-1=0\)
Thus the function is continuous at \(x=1\).
\(f^{\prime}(x)=\left\{\begin{array}{cc}
3 x^{2}, & 1 < x < \infty \\
1, & -\infty < x \leq 1
\end{array}\right.\)
Now, check the differentiability at \(x=1\).
LHD at \(x=1 \Rightarrow 1\)
RHD at \(x=1 \Rightarrow 3(1)^{2}=3\)
\(A s, L H D \neq R H D\), function is not differentiable.