Continuity of a Function

Choose the correct statement on the continuity of the function $f$ given by  $f\left(x\right)=\left\{\begin{array}{ll}x,& \mathrm{if} x\ge 0\\ x^2,& \mathrm{if} x<0\end{array}\right.$at $x=0$

What is non-removable discontinuity. Explain with an example.

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.

 Define infinite type of non-removable discontinuity with one example.

 Define isolated point 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.

Define infinite discontinuity. Find the point of discontinuity of $f\left(x\right)=\frac{1}{x^2}$.

Define infinite discontinuity. Find the point of discontinuity of $f\left(x\right)=\frac{1}{x}$.

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