Continuity and Differentiability

If $f\left(x\right)=\left{\begin{matrix} e^{x}, & x\leq\,0 \\ \left|1-x\right|, & x>0 \end{matrix}\right$, then

Let$f\left(x\right)=\left{\begin{matrix} \begin{matrix} x^{p}\sin\,\frac{1}{x}, & x\neq\,0 \end{matrix} \\ \begin{matrix} 0, & x=0 \end{matrix} \end{matrix}\right$then $f\left(x\right)$ is continuous butnot…

Let $f$ be the function defined by $f(x)=\left\{\begin{array}{cc} \frac{x^2-1}{x^2-2|x-1|-1}, & x \neq 1 \\ \frac{1}{2}, & x=1 \end{array}\right.$

If $f(x)=\left\{\begin{array}{cc}\frac{x^2 \log (\cos x)}{\log (1+x)}, & x \neq 0 \\ 0, & x=0\end{array}\right.$, then at $x=0, f(x)$ is

$f(x)=\left\{\begin{array}{cc}4 & -\infty \lt x \lt -\sqrt{5} \\ x^2-1 & -\sqrt{5} \leq x \leq \sqrt{5} \\ 4 & \sqrt{5} \leq x \lt \infty\end{array}\right.$ If $k$ is the number of points where $f(x)$ is not…

Let $\mathrm{f}$ be the function defined by $f(x)=\left\{\begin{array}{ll}\frac{x^{2}-1}{x^{2}-2|x-1|-1}, & x \neq 1 \\ 1 / 2, & x=1\end{array}\right.$

Let $f(x)={{x}^{p}\sin\frac{1}{x}, x≠00, x=0)}$ then $f(x)$ is continuous but not differentiable at $x=0$, if

Consider the greatest integer function, defined by $f(x)=[x], 0≤x<2$. Then

The value of $f(0)$, so that the function $f(x)=\frac{2x-{\sin}^{-1}x}{2x+{\tan}^{-1}x}$ is continuous at each point in its domain, is

If $f(x)=\left\{\begin{array}{cl}\frac{x \log \cos x}{\log \left(1+x^{2}\right)} & , x \neq 0 \\ 0 & , x=0\end{array}\right.$ then $f(x)$ is

The number of points at which the function $\mathrm{f}(\mathrm{x})=\frac{1}{\log |\mathrm{x}|}$ is discontinuous is:

Let $f: R \rightarrow R$ be a function such that $f(x+y)=f(x)+f(y), \forall x, y \in \mathbb{R}$ If $\mathrm{f}(\mathrm{x})$ is differentiable at $\mathrm{x}=0$, then which one of the following is incorrect?