Discontinuity of a Function

Function $f\left(x\right)=\cos \left(\frac{1}{x}\right)$ has oscillatory discontinuity at point $x=0$.

Function $f\left(x\right)=\sin \left(\frac{1}{x}\right)$ has oscillatory discontinuity at point $x=0$.

Define the oscillatory discontinuity with one example.

This is the graph of a function $h\left(x\right)$.

Find the $x$-value at which $h\left(x\right)$ has an isolated point discontinuity.

This is the graph of a function $h\left(x\right)$.

Find the $x$-value at which $h\left(x\right)$ has a isolated point discontinuity.

This is the graph of a function $f\left(x\right)$. Dashed lines represent asymptotes.

Select the $x$-value at which $f\left(x\right)$ has an isolated point discontinuity.

This is the graph of a function $h\left(x\right)$.

Find the $x$-value at which $h\left(x\right)$ has a isolated point discontinuity.

This is the graph of a function $h\left(x\right)$.

Find the $x$-value at which $h\left(x\right)$ has a missing point discontinuity.

This is the graph of a function $f\left(x\right)$. Dashed lines represent asymptotes.

Select the $x$-value at which $f\left(x\right)$ has a missing point discontinuity.

Let function $f\left(x\right)=\frac{\sin x}{x}$.

The function $f\left(x\right)$ has a missing point discontinuity at $x=0$.

Let function $f\left(x\right)=\frac{x^2-4}{\left(x-2\right)}$.

The function $f\left(x\right)$ has a missing point discontinuity at $x=a$. Find the value of $\text{'}a\text{'}$.

Define the missing point discontinuity with one example.

The function $f\left(x\right)=\frac{4-x^2}{4x-x^3}$ is :

Let $g\left(x\right)=\left\{\begin{array}{c}1-x;\\ x+2;\\ 4-x;\end{array}\right.\begin{array}{c}0\le x\le 1\\ 1<x<2\\ 2\le x\le 4\end{array}$, then the numbers of points where $g\left(g\left(x\right)\right)$ is discontinuous is

Discuss the continuity of the following functions. Which of these functions have removable discontinuity? Redefine such a function at the given point so as to remove discontinuity.

$$\begin{gathered} f\left(x\right)=\frac{\sin (a+x)+\sin (a-x)-2\sin a}{x\sin x} \\ =\sin a \end{gathered}$$ $\left.\begin{array}{l}\text{ for }x\ne a\\ \text{ for }x=a\end{array}\right\}$ at $x=a$

Discuss the continuity of the following functions. Which of these functions have removable discontinuity? Redefine such a function at the given point so as to remove discontinuity.

 $f\left(x\right)=\left\{\begin{array}{l}\frac{\left|x\right|}{x}\\ 1\end{array}\right.\begin{array}{l}\text{ for }x\ne 0\\ \text{ for }x=0\end{array}$ at origin

Discuss the continuity of the function. Is this functions has removable discontinuity ?.  Redefine such a function at the given point so as to remove discontinuity.

$f\left(x\right)=\left\{\begin{array}{l}\frac{{\left(3^{\sin x}-1\right)}^2}{x\log (1+x)} \mathrm{for} x\ne 0\\ 2\log 3 \mathrm{for} x=0\end{array}\right.$,at $x=0$

Discuss the continuity of the function $f\left(x\right)$ defined by 

$f\left(x\right)=\begin{array}{ll}\frac{4^x-e^x}{6^x-1},& \mathrm{for} x\ne 0\\ \log \left(\frac{2}{3}\right),& \mathrm{for} x=0\end{array}\} \mathrm{at} x=0$

If the discontinuity is removable redefine the function, so that it becomes continuous.

Discuss the continuity of the function $f\left(x\right)$ defined by $f\left(x\right)=\begin{array}{ll}\frac{1-\sin x}{{\left(\mathrm{\pi }-2\mathrm{x}\right)}^2},& \mathrm{for} x\ne \frac{\mathrm{\pi }}{2}\\ \frac{2}{7},& \mathrm{for} x=\frac{\mathrm{\pi }}{2}\end{array}\} \mathrm{at} x=\frac{\mathrm{\pi }}{2}$

If the discontinuity is removable redefine the function, so that it becomes continuous.

Discuss the continuity of the function $f\left(x\right)$ defined by $f\left(x\right)=\begin{array}{ll}\frac{\left(e^{2x}-1\right)\tan x}{x \sin x},& \mathrm{for} x\ne 0\\ e^2,& \mathrm{for} x=0\end{array}\} \mathrm{at} x=0$

If the discontinuity is removable redefine the function, so that it becomes continuous.