For the function $f(x)=|x-3|$, which of the following is not correct?

• Correct answer
  A

  The function is not continuous at $x=-3$
• B

  The function is continuous at $x=3$
• C

  The function is differentiable at $x=0$
• D

  The function is differentiable at $x=-3$

Solution

$\begin{gathered} f(x)=|x-3| \\ \Rightarrow \quad f(x)= \begin{cases}(x-3), & x \geq 3 \\ (3-x), & x \lt 3\end{cases} \\ f^{\prime}(x) \text { at } x=3^{+}=1 \\ f^{\prime}(x) \text { at } x=3^{-}=-1 \end{gathered}$ Thus, $f(x)$ is not differentiable at $x=3$ but $f(x)$ is continuous at $x=3$.

Hence, the correct answer is option: 'A'.