If the function $f(x) = \left( \frac{5x - 8}{8 - 3x} \right)^{\frac{3}{2x-4}}$ for $x \neq 2$ is continuous at $x = 2$, then the value of $f(2)$ is...

Discuss the continuity of the function $f,$ where $f$ is defined by $f(x) = \begin{cases} 2x, & \text{if } x < 0 \\ 0, & \text{if } 0 \le x \le 1 \\ 4x, & \text{if } x > 1 \end{cases}$ at $x=3.$

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 $x = 0$.

If $f(x) = \begin{cases} \sin x, & \text{if } x \leq 0 \\ x^2+a^2, & \text{if } 0 < x < 1 \\ bx+2, & \text{if } 1 \leq x \leq 2 \\ 0, & \text{if } x > 2 \end{cases}$ is continuous on $\mathbb{R}$, then $a+b+ab = $

Let the function $f$ be defined by $f(x) = \frac{x - |x|}{x}$ for $x \neq 0$ and $f(0) = 2$. Then $f$ is:

Examine the continuity of $f$,where $f$ is defined by $f(x) = \begin{cases} \sin x - \cos x, & \text{if } x \neq 0 \\ -1, & \text{if } x = 0 \end{cases}$