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:

Let $f\left(x\right)=[n+p\sin x],x\in (0,\pi ),n\in I$and$p$ is a prime number. The number of points where $f\left(x\right)$ is not differentiable is

( Here $\left[x\right]$ represents the greatest integer less than or equal to $x$ )

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

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?

Let $f:R\to R$ be defined as

$$\begin{gathered} f\left(x\right)=\left\{\begin{array}{l}\left[e^x\right]; x<0 \\ a{e}^x+\left[x-1\right]; 0\le x<1 \\ b+\left[\sin \left(\mathrm{\pi x}\right)\right]; 1\le x<2\\ \left[e^{-x}\right]-c; x\ge 2\end{array}\right. \end{gathered}$$

where, $a,b,c\in R$ and $\left[·\right]$ denotes greatest integer function. Then which of the following is true?

If $f\left(x\right)=\left\{\begin{array}{l}x; x\in Q\\ -x; x\notin Q\end{array}\right.$, find the points where $f\left(x\right)$ is continuous.

Find the number of discontinuity of the function $f\left(x\right)=\left[5x\right]+\left\{3x\right\}$ in $\left[0,5\right]$ where $\left[y\right] \& \left\{y\right\}$ denotes largest integer less than or equal to $y$ and fractional part of $y$ respectively.

Let $f\left(x\right)=\underset{n\to \infty }{\lim }\left[\frac{\ln \left(1+x\right)-x^{2n}\sin x^2}{1+x^{2n}}\right]$. Which of the following statements(s) is(are) correct?

If $f\left(x\right)=\underset{n\to \infty }{\lim }\left[\frac{\left\{1-\cos \left(1-\tan \left({\displaystyle \frac{\mathrm{\pi }}{4}}-x\right)\right)\right\}{\left(x+1\right)}^n+\lambda \sin \left(n-\sqrt{n^2-8n}\right)x}{x^2{\left(x+1\right)}^n+x}\right]; x\ne 0$ is continuous at $x=0$, then find the value of $f\left(0\right)+4\lambda$.

If the function $f$ defined by $f\left(x\right)=\left\{\begin{array}{l}\frac{5^x-5^{-x}}{2}; x\ne 0\\ k; x=0\end{array}\right.$ is continuous at $x=0$, then $k=$

Suppose a function $f\left(x\right)$ is defined by

$$\begin{gathered} f\left(x\right)=\left\{\begin{array}{l}{x}^2-1; -1\le x<0 \\ 2x; 0<x<1 \\ 1; x=1 \\ -2x+4; 1<x<2\\ 0; 2<x<3\end{array}\right. \end{gathered}$$

Then answer the following questions:

$\left(1\right)$ Does $f\left(-1\right)$ exists?

$\left(2\right)$ Does $\underset{x\to -1^{+}}{\lim }f\left(x\right)$ exists?

$\left(3\right)$ Does $\underset{x\to -1^{+}}{\lim }f\left(x\right)=f\left(-1\right)$?

$\left(4\right)$ If $f$ defined at $x=3$?

$\left(5\right)$ Is $f$ continuous at $x=3$?

$\left(6\right)$ At what value of $x$ is $f$ continuous?

$\left(7\right)$ What value should be assigned to $f\left(2\right)$ to make the extended function continuous at $x=2$?

A function $f\left(x\right)$ is defined as follows:

$f\left(x\right)=\left\{\begin{array}{l}\frac{\sin x}{x}; x\ne 0\\ 2; x=0\end{array}\right.$

Is $f\left(x\right)$ continuous at $x=0$?

If not, redefine it so $f\left(x\right)$ becomes continuous at $x=0$.

$f\left(x\right)=\left[\sin x\right]$, where $\left[·\right]$ denotes the greatest integer function, is/are continuous at

Test the continuity of $f\left(x\right)$ at $x=0$ where $f\left(x\right)=\left\{\begin{array}{l}\begin{array}{cc}\frac{x{e}^{\frac{1}{x}}}{1+e^{{\displaystyle \frac{1}{x}}}}; & x\ne 0\end{array}\\ \begin{array}{cc}0; & x=0\end{array}\end{array}\right.$.

Consider  $f\left(x\right)=\left\{\begin{array}{l}x{\left[x\right]}^2{\log }_{1+x}\left(2\right) \text{for} -1<x<0\\ \frac{\ln \left(e^{x^2}+2\sqrt{\left\{x\right\}}\right)}{\tan \sqrt{x}} \text{for} 0<x<1\end{array}\right.$where $\left[·\right]$ and $\left\{·\right\}$ are the greatest integer function and fractional part function respectively, then