Question

Let $f\left(x\right)=\{\begin{array}{cc}{x}^2/2& 0<x\le 1\\ 2x^2-3x+3,& 1<x<2\end{array}$ then which is incorrect

• A. $f$ is continuous in $(0,2)$
• B. $f$ is not continuous at all points in $(0,2)$
• C. $f$ is differentiable in $(0,2)$
• D. $L{t}_{x\to 1^{+}}f\left(x\right)=L{t}_{x\to 1^{-}}f\left(x\right)$

Answer 1

Answer: (A), (C), (D)

$f$ is continuous in $(0,2)$
$f$ is differentiable in $(0,2)$
$L{t}_{x\to 1^{+}}f\left(x\right)=L{t}_{x\to 1^{-}}f\left(x\right)$

The point where we need to check the continuity and differentiability is $x=1$.
We have,
${lim}_{x\to 1^{-}}f\left(x\right)=\frac{x^2}{2}=\frac{1}{2}{lim}_{x\to 1^{+}}f\left(x\right)={2x}^2-3x+3=2$
As the left hand and right hand limits are not equal, the function is not continuous at $x=1$. Hence, it is not differentiable at $x=1$.
Hence, options A, B and D all represent incorrect statements.