Find the values of $k$ so that the function $f$ is continuous at the indicated point. $f(x) = \begin{cases} kx + 1, & \text{if } x \le 5 \\ 3x - 5, & \text{if } x > 5 \end{cases}$ at $x = 5$. (in $/5$)

Which of the following statements is true for the graph of the function $f(x) = \log x$?

Let $f, g: R \to R$ be two functions defined by $f(x) = \begin{cases} x \sin \left( \frac{1}{x} \right), & x \ne 0 \\ 0, & x = 0 \end{cases}$ and $g(x) = x f(x)$.
Statement $I$: $f$ is a continuous function at $x = 0$.
Statement $II$: $g$ is a differentiable function at $x = 0$.

Show that the function defined by $f(x)=\cos \left(x^{2}\right)$ is a continuous function.

If $f(x) = \begin{cases} (1 + 2x)^{1/x}, & x \ne 0 \\ e^2, & x = 0 \end{cases}$,then:

The values of $a$ and $b$ for which the function $f(x) = \begin{cases} 1+|\sin x|^{a/|\sin x|}, & -\pi / 6 < x < 0 \\ b, & x = 0 \\ e^{\tan 2 x / \tan 3 x}, & 0 < x < \pi / 6 \end{cases}$ is continuous at $x = 0$ are