The set of values of $x$ for which the function $f(x) = \log \left(\frac{x-1}{x+2}\right)$ is continuous, is

In order that the function $f(x) = (x + 1)^{1/x}$ is continuous at $x = 0$,$f(0)$ must be defined as

The value of $p$ for which the function $f(x) = \begin{cases} \frac{(4^x - 1)^3}{\sin(\frac{x}{p}) \log(1 + \frac{x^2}{3})}, & x \ne 0 \\ 12(\log 4)^3, & x = 0 \end{cases}$ is continuous at $x = 0$ is:

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

If $f(x) = \frac{x - e^x + \cos 2x}{x^2}$ for $x \neq 0$ is continuous at $x = 0$,then which of the following is true? (Note: $[x]$ and $\{x\}$ denote the greatest integer and fractional part functions,respectively.)

Discuss the continuity of the function $f$ given by $f(x) = |x|$ at $x = 0$.