Question

A function is defined as follows:
$f\left(x\right)=\{\begin{array}{c}{x}^3;x^2<1\\ x;x^2\ge 1\end{array}$ discuss the continuity and differentiability at $x=1$

• A. Continuous but not differentiable at $x=1$
• B. Continuous and differentiable at $x=1$
• C. not Continuous and not differentiable at $x=1$
• D. none of these

Answer 1

The correct option is A Continuous but not differentiable at $x=1$

$f\left(1^{-}\right)=\underset{h\to 0}{lim}\left({1-h}^3\right)=1$
$f\left(1^{+}\right)=\underset{h\to 0}{lim}(1-h)=1$
$f^{\prime }\left(1^{-}\right)=\frac{d}{dx}\left(x^3\right)=3(1{)}^2=3$
$f^{\prime }\left(1^{+}\right)=\frac{d}{dx}\left(x\right)=1$
So, function is continuous but not differentiable at $x=1$