Is the function defined by $f(x) = |x|$ a continuous function?

If $f(x) = \begin{cases} \frac{x^3 + x^2 - 16x + 20}{(x-2)^2}, x \neq 2 \\ k, x = 2 \end{cases}$ is continuous at $x = 2$, then $k = \rule{1cm}{0.15mm}$

If $f(x) = \left[\tan \left(\frac{\pi}{4} + x\right)\right]^{\frac{1}{x}}$ for $x \neq 0$ and $f(x) = k$ for $x = 0$,is continuous at $x = 0$,then $k =$

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.)

If $f(x) = \begin{cases} \frac{a \sin x - b x + c x^2 + x^3}{2 \log(1+x) - 2x + x^2 - \frac{2}{3}x^3} &, x \neq 0 \\ 0 &, x=0 \end{cases}$ is continuous at $x=0$,then find the relation between $a, b, c$.

Find all points of discontinuity of $f$,where $f$ is defined by $f(x) = \begin{cases} \frac{x}{|x|}, & \text{if } x < 0 \\ -1, & \text{if } x \ge 0 \end{cases}$. Is $f$ a continuous function?