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

If $f(x) = \begin{cases} \frac{a|x|+x^2-2(\sin |x|)(\cos |x|)}{x} & , x \neq 0 \\ b & , x=0 \end{cases}$ is continuous at $x=0$, then $a+b$ is equal to:

The points at which the function $f(x) = \frac{x + 1}{x^2 + x - 12}$ is discontinuous,are

Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined as $f(x) = \begin{cases} 5, & \text{if } x \le 1 \\ a + bx, & \text{if } 1 < x < 3 \\ b + 5x, & \text{if } 3 \le x < 5 \\ 30, & \text{if } x \ge 5 \end{cases}$. Then $f$ is

If $f(x) = \begin{cases} 1+6x-3x^2, & x \leq 1 \\ x+\log_2(b^2+7), & x > 1 \end{cases}$ is continuous at all real $x$,then $b=$

Find the relationship between $a$ and $b$ so that the function $f$ defined by $f(x) = \begin{cases} ax + 1, & \text{if } x \le 3 \\ bx + 3, & \text{if } x > 3 \end{cases}$ is continuous at $x = 3$.