Question

At $x=1$, the function $f(n) = \begin{cases} x^{3} -1 & 1 < x > \infty \\ x-1, & \infty < x \ge 1 \end{cases}$ is

• A. continuous and differentiable
• B. continuous and non-differentiable
• C. discontinuous and differentiable
• D. discontinuous and non-differentiable

Answer 1

Answer: (B)

$\displaystyle\lim _{x \rightarrow 1+} x^{3}-1=0$
$\displaystyle\lim _{x \rightarrow 1-}(x-1)=0$
$F$ is continuous
$f(n) = \begin{cases} 3x^2 & 1 < x < \infty \\ 1 & -\infty < x < \end{cases}$
$f'\left(1^{+}\right)=3, f'\left(1^{-}\right)=1$
$\Rightarrow f$ is not differentiable