Question

Let a function $f(x)$ be defined by $f(x)=\frac{x-|x-1|}{x}$, then which of the following is not true?

• A. Discontinuous at $x=0$
• B. Discontinuous at $x=1$
• C. Not differentiable at $x=0$
• D. Not differentiable at $x=1$

Answer 1

Answer: (B)

We have
$f(x)=\frac{x-|x-1|}{x}=\{\begin{array}{ll}\frac{x+x+1}{x},& \text{x<1,x=0 }\\ \frac{x-(x-1)}{x},& \text{if x≥1 }\end{array}$
$=\{\begin{array}{ll}\frac{2x-1}{x},& \text{x<1,x=0 }\\ \frac{1}{x},& \text{x≥1 }\end{array}$
Clearly, $f(x)$ is discontinuous at $x=0$ as it is not defined at $x=0.$ Since $f(x)$ is not defined at $x=0,f(x)$ cannot be differentiable at $x=0$. Clearly, $f(x)$ is continuous at $x=1$, but it is not differentiable at $x=1$, because $L{f}^{\prime }(1)$ and $R{f}^{\prime }(1)=1$