Q. Find the relationship between $a$ and $b$, so that the function $f\left(x\right)$ defined by $f\left(x\right)=$ $\{\begin{array}{c}ax+2,ifx\le 4\\ bx+4,ifx>4\end{array}$
is continuous at $x=3$

Q. Find all points of discontinuity of $f$, where $f$ is defined by
$f\left(x\right)=$ $\{\begin{array}{c}|x|+3=-x+3,\text{if}x\le -3\\ -2x,\text{if}-3<x<3\\ 6x+2,\text{if}x\ge 3\end{array}$

Q. Examine the continuity of the function $f\left(x\right)=2x^2-1$ at $x=3$.

Q. Is the function $f$ defined by
$f\left(x\right)=$ $\{\begin{array}{c}x,ifx\le 1\\ 5,ifx>1\end{array}$
continuous at $x=0$? At $x=1$? At $x=2$?

Q. Find all points of discontinuity of $f$, where $f$ is defined by
$f\left(x\right)=$ $\{\begin{array}{c}{x}^{10}-1,\text{if}x\le 1\\ x^2,\text{if}x>1\end{array}$

Q. Show that the function defined by $g\left(x\right)=x-\left[x\right]$ is discontinuous at all integral points. Here $\left[x\right]$ denotes the greatest integer less than or equal to $x$.

Q. Prove that the function $f\left(x\right)=x^n$ is continuous at $x=n$, where $n$ is a positive integer.

Q. (i) $f\left(x\right)=x-5$ (ii) $f\left(x\right)=$ $\frac{1}{x-5}$, $x\ne 5$

Q. Find all points of discontinuity of $f$, where $f$ is defined by
f(x) = $\{\begin{array}{c}\frac{x}{|x|}\text{if}x<0\\ -1\text{if}x\ge 0\end{array}$

Q. Find all points of discontinuity of $f$, where $f$ is defined by $f\left(x\right)=$ $\{\begin{array}{c}\frac{|x|}{x},\text{if}x\ne 0\\ 0,\text{if}x=0\end{array}$

Q. Discuss the continuity of the function $f$, where $f$ is defined by
$f\left(x\right)=$ $\{\begin{array}{c}3,if0\le x\le 1\\ 4,if1<x<3\\ 5,if3\le x\le 10\end{array}$

Q. Is the function defined by
$f\left(x\right)=$ $\{\begin{array}{c}x+5,ifx\le 1\\ x-5,ifx>1\end{array}$
a continuous function ?

Q. Prove that the function $f\left(x\right)=5x-3$ is continuous at $x=0$, at $x=-3$ and at $x=5$.

Q. Find all points of discontinuity of $f$, where $f$ is defined by
f(x) = $\{\begin{array}{c}x+1,\text{if}x\ge 1\\ x^2+1,\text{if}x<1\end{array}$

Q. Find all points of discontinuity of $f$, where $f$ is defined by
$f\left(x\right)=$ $\{\begin{array}{c}2x+3,ifx\le 2\\ 2x-3,ifx>2\end{array}$

Q. Find all points of discontinuity of $f$, where $f$ is defined by
$f\left(x\right)=$ $\{\begin{array}{c}{x}^3-3,ifx\le 2\\ x^2+1,ifx>2\end{array}$

Q. Discuss the continuity of the function f, where f is defined by
f(x) = $\{\begin{array}{c}2x,ifx<0\\ 0,if0\le x\le 1\\ 4x,ifx>1\end{array}$

Q. Discuss the continuity of the function$f$, where $f$ is defined by
$f\left(x\right)=$ $\{\begin{array}{c}-2,ifx\le -1\\ 2x,if-1<x\le 1\\ 2,ifx>1\end{array}$

Q. $f\left(x\right)=$ $\frac{x^2-25}{x+5}$, $x$ $\ne$ $5$

Q. Discuss the continuity of the function defined by $f\left(x\right)=|x-5|$.