I A Maron Solutions for Chapter: Introduction to Mathematical Analysis, Exercise 14: Continuity of a Function. Points of Discontinuity and Their Classification

For each of the following functions find the points of discontinuity and determine the jumps of the function at these points :

$f\left(x\right)=\frac{4}{x^2-2x+1}$

For each of the following functions find the points of discontinuity and determine the jumps of the function at these points :

$f\left(x\right)=x+\frac{x+2}{\left|x+2\right|}$

For each of the following functions find the points of discontinuity and determine the jumps of the function at these points :

$f\left(x\right)=\frac{2\left|x-1\right|}{x^2-x^3}$

For each of the following functions find the points of discontinuity and determine the jumps of the function at these points :

$f\left(x\right)=\left\{\begin{array}{l}-x \mathrm{for} x\le 1\\ \frac{2}{x-1} \mathrm{for} x>1\end{array}\right.$

Redefine the following functions at the point $x=0$ so as to make them continuous :

$f\left(x\right)=\frac{\tan x}{x}$

Redefine the following functions at the point $x=0$ so as to make them continuous :

$f\left(x\right)=\frac{5x^2-3x}{2x}$

Redefine the following functions at the point $x=0$ so as to make them continuous :

$f\left(x\right)=\frac{\sqrt{1+x}-1}{x}$

Redefine the following functions at the point $x=0$ so as to make them continuous :

$f\left(x\right)=\frac{{\sin }^{2}x}{1-\cos x}$