Question

If $f\left(x\right)=x-2,x<3$
$=2x-5,x\ge 3$, then.

• A. function is not continuous at $x=3$
• B. function is continuous and differentiable at $x=3$
• C. function is differentiable at $x=3$
• D. function is not continuous but differentiable at $x=3$

Answer 1

Answer: (A)

function is not continuous at $x=3$

$f\left(x\right)=\{\begin{array}{c}x-2;x<3\\ 2x-5;x\ge 3\end{array}$

at $x=3$

$\underset{x\to 3^{-}}{lim}f\left(x\right)=\underset{x\to 3^{-}}{lim}(x-2)=1$

$\underset{x\to 3^{+-}}{lim}f\left(x\right)=\underset{x\to 3^{+}}{lim}(2x-5)=6-5=1$

$f\left(3\right)=6-5=1$

LHL $=$ RHL $=f\left(3\right)$

at $x=3$

$f^{\prime }\left(x\right)=\{\begin{array}{c}1;x<3\\ 2;x\ge 3\end{array}$

$\underset{x\to 3^{-}}{lim}{f}^{\prime }\left(x\right)=1\&\underset{x\to 3^{+}}{lim}{f}^{\prime }\left(x\right)=2$

LHL$\ne$RHL $\Rightarrow$Not differentiable