Show that: $f\left(x\right)=\left\{\begin{array}{c}x+1,\text{ if }x\ge 1\\ x^2+1,\text{ if }x<1\end{array}\right.$ is continuous at $x=1$.

Show that: $f\left(x\right)=\left\{\begin{array}{l}{x}^3-3,\text{ if }x\le 2\\ x^2+1,\text{ if }x>2\end{array}\right.$ is continuous at $x=2$.

Find the values of $a$ and $b$ such that the following function is continuous. $f\left(x\right)=\left\{\begin{array}{l}5,\text{ when }x\le 2\\ ax+b,\text{ when }2<x<10\\ 21,\text{ when }x\ge 10\end{array}\right.$.

Find the values of $a$ for which the following functions $ƒ$, defined as $f\left(x\right)=\left\{\begin{array}{l}a\sin \frac{\pi }{2}(x+1),x\le 0\\ \frac{\tan x-\sin x}{x^3},x>0\end{array}\right.$ is continuous at $x=0$.

Prove that the function $f$ given by $f\left(x\right)=|x-3|, x\in R$ is continuous but not differentiable at $x=3$.

Show that function $f\left(x\right)=\left\{\begin{array}{ll}(7x+5),& \text{ when }x\ge 0\\ (5-3x),& \text{ when }x<0\end{array}\right.$ is a continuous function.

Show that function $f\left(x\right)=\left\{\begin{array}{r}\sin x,\text{ if }x<0\\ x,\text{ if }x\ge 0\end{array}\right.$ is continuous.

Show that function $f\left(x\right)=\left\{\begin{array}{cc}\frac{x^n-1}{x-1},& \text{ when }x\ne 1\\ n,& \text{ when }x=1\end{array}\right.$ is continuous.

Discuss the continuity of the function $f\left(x\right)=\sec x$.