Types of Discontinuity

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.

If [.] denotes the Greatest integer function, then $f\left(x\right)=[x{]}^2-\left[x^2\right]$ is discontinuous at

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

A point in the domain that cannot be filled in so that the resulting function is continuous is called

If $f\left(x\right)=\frac{1}{1-x}$, the number of points of discontinuity of $f\left\{f\left[f\left(x\right)\right]\right\}$ is

Let $f\left(x\right)=\left\{\begin{array}{ll}{x}^2-1,& x\in (-\infty ,0)\\ min\left(x,x^2\right),& x\in [0,\infty )\end{array}\right.$If $m$ and $n$ represents number of points, where $f\left(x\right)$ is discontinuous and non differentiable respectively, then $(m+n)$ equals

If $x\in R^{+}$ and $n\in N$, we can uniquely write $x=mn+r$ where $m\in W$ and $0\le r<n$. We define $x mod n=r$ for example $10.3 mod 3=1.3$. The number of points of discontinuity of the function $f\left(x\right)=(x mod 2{)}^2+(x mod 4)$ in the interval $0<x<9$ is

Consider $f\left(x\right)=\sin x\forall x\in \left[0,\frac{\pi }{2}\right];f\left(x\right)+f(\pi -x)=2\forall x\in \left(\frac{\pi }{2},\pi \right]$ and $f\left(x\right)=f(2\pi -x)\forall x \in (\pi ,2\pi ).$  If $n,m$ denotes number of points where $f\left(x\right)$ is discontinuous and non-differentiable respectively in $[0,2\pi )$, then value of $n+m$ is