Limits, Continuity and Differentiability

For real numbers $a$ and $b$, consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ given by$f(x) = \begin{cases} -ax - b & \text{if } x < -1, \\ 5x + 1 & \text{if } -1 \leq x \leq 1, \\ a^2x + 3b & \text{if } x > 1. \end{cases}$How...

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be the function given by$f(x) = |x - 2| + 3|x - 1| + ||x - 2| - 1|$.What is the number of points where $f$ is NOT differentiable?

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)=\left|x^3-3 x\right|[x]$, where $[x]$ denotes the greatest integer less than or equal to $x$. Which one of the following statements is TRUE?

Let $f, g: R \rightarrow R$ be functions. If $g$ is continuous, then which one of the following cases implies that $f$ is continuous?

Which one of the following functions is differentiable at $x=0$ ?

If $f(x)=a x^2+b x+c$ and $f\left(\frac{1}{n}\right)=\frac{n+1}{n^2}$ for all $n \in N$, then what is the value of $\lim \limits_{x \rightarrow 0} f^{\prime}(x)$ ?