Continuity of a Function

Let $g\left(x\right)$  be a polynomial of degree one and $f\left(x\right)$  be a continuous and differentiable function defined by $f\left(x\right)=\left\{\begin{array}{cc}g\left(x\right),& x\le 0\\ {\left(\frac{1+x}{2+x}\right)}^{\frac{1}{x}},& x>0\end{array}\right.$. If $f^{\text{'}}\left(1\right)=f^{\text{'}}\left(-1\right)$, then

$f\left(x\right)={\left({\sin }^{-1}x\right)}^2·\cos \left(1/x\right)$ if $x\ne 0; f\left(0\right)=0, f\left(x\right)$ is:

$\text{If }\text{f}\left(x\right)=[\begin{array}{cc}x+\left\{x\right\}+x\text{sin}\left\{x\right\}& \text{for}x\ne 0\\ 0& \text{for}x=0\end{array}\text{where}\left\{x\right\}\text{denotes the fractional part function, then:}$

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 $x=0$

Let f be a real-valued function defined on the interval  $(0,\infty )$  by  $f(x)=\ell nx+{\displaystyle \underset{0}{\overset{x}{\int }}\sqrt{1+\sin t}dt.}$  Then which of the following statement(s) is (are) true?

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 $f\left(x\right)$.

Find the $x$-value at which $f\left(x\right)$ has a jump discontinuity.

 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 $x=0$.