Question

Let f(x) = ${\left|x\right|}^x$ if $x\ne 0$
f(x) = 1 if x = 0 Which of the following statement(s) is/are correct?

• A. f is continuous at x = 0
• B. f is not differentiable at x = 0
• C. f is derivable at x = 0
• D. f is discontinuous at x = 0

Answer 1

Answer: (A), (B)

The correct options are
A f is continuous at x = 0
B f is not differentiable at x = 0

R.H.L$=\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}{x}^x$
=$e^{{lim}_{x\to 0^{+}}x\ln x}=e^{{lim}_{x\to 0^{+}}(-x)}=1$
L.H.L$=\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{-}}{lim}{(-x)}^x$
= $e^{{lim}_{x\to 0^{-}}x\ln (-x)}=e^{{lim}_{x\to 0^{-}}-x}=1$
$\therefore$ f(x) is continuous at x = 0
$f^{\prime }\left(0^{+}\right)=\underset{h\to 0^{+}}{lim}\frac{h^h-1}{h}$
= $\underset{h\to 0^{+}}{lim}{h}^h[1+\iota nh]=$ not defined
$\Rightarrow$ f is not differentiable at $x=0$