Continuity

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 $x=0$

Find the constants $a$ and $b$ so that the function $f$ defined below is continuous in $R$

 $f\left(x\right)=\left\{\begin{array}{ll}1,& x⩽3\\ ax+b,& 3<x<5\\ 7,& x⩾5\end{array}\right.$

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 $f\left(x\right)=\frac{x}{1-x}$ has infinite discontinuity at $x=1$.

Define infinite discontinuity with one example.

Let the graph shown below:

Determine the domain and range of the given graph. Classify is as discrete or continuous. 

Discrete functions are always continuous. 

Give one example of discrete function which proves that discrete functions are not continuous. 

State whether discrete functions are continuous or discontinuous. Find derivative of following discrete function.

$f\left(x\right)=\left\{\begin{array}{l}5,x=1\\ 2,x=2\\ 1,x=5\end{array}\right.$

State whether discrete functions are continuous or discontinuous. Find derivative of following discrete function.

$f\left(x\right)=\left\{\begin{array}{l}a,x=1\\ b,x=2\\ c,x=5\end{array}\right.$

State whether discrete functions are continuous or discontinuous. Find derivative of following discrete function.

$f\left(x\right)=\left\{\begin{array}{l}2,x=1\\ 3,x=2\\ 4,x=5\end{array}\right.$

State whether discrete functions are continuous or discontinuous. Find derivative of following discrete function.

$f\left(x\right)=\left\{\begin{array}{l}3,x=1\\ 4,x=2\\ 6,x=5\end{array}\right.$

State whether discrete functions are continuous or discontinuous. Find derivative of following discrete function.

$f\left(x\right)=\left\{\begin{array}{l}1,x=1\\ 2,x=2\\ 5,x=5\end{array}\right.$

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

Define infinite discontinuity. Find the point of discontinuity of $f\left(x\right)=\frac{1}{2x-3}$.