$A$ function $f(x)$ is defined as $f(x) = \begin{cases} x^m \sin \frac{1}{x} & x \neq 0, m \in N \\ 0 & x = 0 \end{cases}$. The least value of $m$ for which $f'(x)$ is continuous at $x = 0$ is

Find the values of $k$ so that the function $f$ is continuous at the indicated point. $f(x) = \begin{cases} kx^2, & \text{if } x \le 2 \\ 3, & \text{if } x > 2 \end{cases}$ at $x=2$.

Let $f$ be a continuous,periodic even function defined on $\mathbb{R}$ such that $f(0) = 1$,$f(2) = -1$ and the period of $f$ is $4$. The minimum number of roots of the equation $f(x) = 0$ in the interval $[-10, 10]$ will be:

If $f(x) = \begin{cases} ax + 7, & \text{if } x < 1 \\ 3x - 1, & \text{if } x = 1 \\ \frac{x + 3}{b}, & \text{if } x > 1 \end{cases}$ is continuous at $x = 1$, then

Examine the following function for continuity: $f(x) = x - 5$.

If $f(x) = \begin{cases} \frac{\sqrt{1+ax}-\sqrt{1-ax}}{x}, & -1 \leq x < 0 \\ \frac{x^2+2}{x-2}, & 0 \leq x \leq 1 \end{cases}$ is continuous on $[-1,1]$, then $a=$