Continuous and Discontinuous Functions

Find the value of $p$ and $q$ for which the function

$f\left(x\right)=\left\{\begin{array}{c}\frac{\sin \left(p+1\right)x\left(\sin x\right)}{x}, x<0\\ q x=0\\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{{\displaystyle \frac{3}{2}}}}, x>0\end{array}\right.$

is continuous for all $x$ in $R$,

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

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 $f\left(x\right)=\left\{\begin{array}{cl}x+6,& x⩽1\\ px+q,& 1<x<2\\ 5x,& x⩾2\end{array}\right.$ and $f\left(x\right)$ is continuous

What is the value of $q$?

Let $f\left(x\right)=\left\{\begin{array}{cl}x+6,& x⩽1\\ px+q,& 1<x<2\\ 5x,& x⩾2\end{array}\right.$ and $f\left(x\right)$ is continuous

What is the value of $p$?

Test the differentiability and continuity of the following function at $x=0$;

$f\left(x\right)=\left\{\begin{array}{cc}\frac{1-e^{-x}}{x},& x\ne 0\\ 1,& x=0\end{array}\right.$.

Let $f\left(x\right)=\left[x^2-x\right]+\left|-x+\left[x\right]\right|$, where $x\in \mathrm{ℝ}$ and $\left[t\right]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is

Let $f\left(x\right)=\left\{\begin{array}{cc}\frac{1}{2}& , x<\frac{\mathrm{\pi }}{2}\\ p& , x=\frac{\mathrm{\pi }}{2}\\ \frac{q\left(1-\sin x\right)}{{\left(\mathrm{\pi }-2\mathrm{x}\right)}^2}& , x>\frac{\mathrm{\pi }}{2}\end{array}\begin{array}{}\\ \end{array}\right.$, if $\underset{x\to \mathrm{\pi }/2}{\lim }f\left(x\right)=f\left(\frac{\mathrm{\pi }}{2}\right)$. Find the value of $p$ and $q$.

If $f\left(x\right)=\left\{x^2\right\}-{\left(\left\{x\right\}\right)}^2$, where $\left\{x\right\}$ denotes the fractional part of $x$, then

If $f\left(x\right)=\left\{\begin{array}{l}\frac{x^3+x^2-16x+20}{{\left(x-2\right)}^2}; x\ne 2\\ k; x=2\end{array}\right.$ at $x=2$ is continuous, then

Find the value of k for which the function 

$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{ll}\frac{\mathrm{sinx}-\mathrm{cosx}}{4\mathrm{x}-\mathrm{\pi }},& \text{if} \mathrm{x}\ne \frac{\mathrm{\pi }}{4}\\ \mathrm{k},& \text{if} \mathrm{x}=\frac{\mathrm{\pi }}{4}\end{array}\right.$

is continuous at $\mathrm{x}=\frac{\mathrm{\pi }}{4}$ 

For what value of $k$ is the function $f$ continuous at $x=0$?

$f\left(x\right)=\left\{\begin{array}{ll}\frac{\sin 2x}{8x},& \mathrm{if} x\ne 0\\ k,& \mathrm{if} x=0\end{array}\right.$

Show that the function $f\left(x\right)={\left(x-a\right)}^2{\left(x-b\right)}^2+x$ takes the value $\frac{a+b}{2}$ for some $x\in \left[a,b\right]$.

If $f\left(a\right)=a, f\left(b\right)=b$, then show that by intermediate value theorem there exists $c\in \left(a,b\right)$ such that $f\left(c\right)=\frac{a+b}{2}$.

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.

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 $k=$

If $f\left(x\right)=\frac{\tan \left({\displaystyle \frac{x}{2}}\right)}{4x},x\ne 0$ is continuous at $=0,$ then $f\left(0\right)=$