Question

If $f(x)=\begin{cases}\frac{x}{|x|}when~x\ne0\\ |x|when~x=0\end{cases}$ is a real valued function, then}

• A. f is continuous but not differentiable at $x=0$
• B. f is both continuous and differentiable at $x=0$
• C. f is not defined at $x=0$
• D. f is neither continuous nor differentiable at $x=0$ \bigskip

Hint:

Discontinuity automatically implies non-differentiability.

Answer 1

The Correct Option is D

Concept: Check continuity first, then differentiability.

Step 1: Behavior for $x>0$.
\[ f(x)=1 \]

Step 2: Behavior for $x<0$.
\[ f(x)=-1 \]

Step 3: Value at $x=0$.
\[ f(0)=0 \]

Step 4: Check left and right limits.
\[ lim_{x\to 0^-}f(x)=-1,\quad lim_{x\to 0^+}f(x)=1 \] Since LHL $\ne$ RHL: function is discontinuous.

Step 5: Conclude differentiability.
Discontinuous $\Rightarrow$ not differentiable.