Question

The function $ f(x)=\left\{ \begin{matrix} |x-3|, & if & x\ge 1 \\ \frac{{{x}^{2}}}{4}-\frac{3x}{2}+\frac{13}{4}, & if & x<1 \\ \end{matrix} \right. $ is

• A. continuous and differentiable at $ x=3 $
• B. continuous at $ x=3, $ but not differentiable at $ x=3 $
• C. continuous and differentiable everywhere
• D. continuous at $ x=1, $ but not differentiable at $ x=1, $

Answer 1

Answer: (B)

Clearly, $ f(x) $ is not differentiable at $ x=3. $
Now, $ \underset{x\to {{3}^{-}}}{\mathop{\lim }}\,\,\,\,\,f(x)=\underset{h\to 0}{\mathop{\lim }}\,\,f(3-h) $
$ =\underset{h\to 0}{\mathop{\lim }}\,|3-h-3| $
$ \underset{x\to {{3}^{+}}}{\mathop{\lim }}\,\,\,f(x)=\underset{h\to 0}{\mathop{\lim }}\,f(3+h) $
$ =\underset{h\to 0}{\mathop{\lim }}\,\,\,|3+h-3|=0 $
and $ f(3)=|3-3|=0 $
$ \therefore $ $ f(x) $ is continuous at $ x=3. $