Question

If $f$ is defined by $f\left(x\right)=\{\begin{array}{ccc}x,& for& 0\le x<1\\ 2-x& for& x\ge 1\end{array}$, then at $x=1$, is:

• A. continuous and differentiable
• B. continuous but not differentiable
• C. discontinuous but differentiable
• D. neither continuous nor differentiable

Answer 1

Answer: (B)

continuous but not differentiable

F is continuous but not differentable

$\underset{x\to 1^{-}}{lim}f\left(x\right)=1$

$\underset{x\to 1^{+}}{lim}f\left(x\right)=2-1=1$

$f\left(1\right)=1$

Hence $f\left(1\right)$ is continuous.

$\underset{x\to 1^{-}}{lim}\frac{f\left(x\right)-f\left(1\right)}{x-1}=\frac{1-h-1}{1-h-1}=1$

$\underset{x\to 1^{+}}{lim}\frac{f\left(x\right)-f\left(1\right)}{x-1}=\frac{2-1-h-1}{1+h-1}=\frac{-h}{h}=-1$

Hence it is not differentiable.