Let $m$ and $n$ be the number of points at which the function $f(x) = \max \{x, x^3, x^5, \dots, x^{21}\}$,$x \in R$,is not differentiable and not continuous,respectively. Then $m + n$ is equal to . . . . . . .

Let $[t]$ represent the greatest integer not exceeding $t$ and $C=1-2e^2$. If the function $f(x)=\begin{cases} [e^x], & x < 0 \\ ae^x+[x-2], & 0 \leq x < 2 \\ [e^{-x}]-C, & x \geq 2 \end{cases}$ is continuous at $x=2$,then $f(x)$ is discontinuous at

Let $R$ be the set of all real numbers and $\alpha \in R$ be positive. Define a function $f: R \rightarrow R$ by $f(0)=0$ and $f(x)=|x|^\alpha \sum \limits_{n=0}^{\infty}\left(1+x^2\right)^{-n}$,for $x \neq 0$. Then the set of real numbers $\alpha$ for which $f$ is continuous at $x = 0$ has

If $[x]$ denotes the greatest integer not exceeding $x$ and if the function $f$ defined by $f(x)= \begin{cases} \frac{a+2 \cos x}{x^2} & , x < 0 \\ b \tan \frac{\pi}{[x+4]} & , x \geq 0 \end{cases}$ is continuous at $x=0$, then the ordered pair $(a, b)$ is equal to

Find all points of discontinuity of $f,$ where $f$ is defined by $f(x) = \begin{cases} x^3 - 3, & \text{if } x \le 2 \\ x^2 + 1, & \text{if } x > 2 \end{cases}$

If $f(x) = \frac{\ln(e^{x^2} + 2\sqrt{x})}{\sqrt{x}}$ is continuous at $x = 0$,then $f(0)$ must be equal to: