EXERCISE 5.1

Find the values of $k$ so that the function $f$ is continuous at the indicated point.
$f\left(x\right)=\left\{\begin{array}{c}k{x}^2, \text{If} x\le 2\\ 3, \text{If} x>2\end{array}\right.\mathrm{a}\mathrm{t} x=2$.

Find the values of $k$ so that the function $f$ is continuous at the indicated point.
$f\left(x\right)=\left\{\begin{array}{c}kx+1, \text{If} x\le \pi \\ \cos x, \text{If} x>\pi \end{array}\right.\mathrm{a}\mathrm{t} x=\pi$.

Find the values of $k$ so that the function $f$ is continuous at the indicated point.

$f\left(x\right)=\left\{\begin{array}{c}kx+1, \text{If} x\le 5\\ 3x-5, \text{If} x>5\end{array}\right.\mathrm{a}\mathrm{t} x=5$

Find the values of $a$ and $b$ such that the function defined by
$f\left(x\right)=\left\{\begin{array}{c}\begin{array}{c}5, \text{If} x\le 2\\ ax+b, \text{If} 2<x<10\end{array}\\ 21, \text{If} x\ge 10\end{array}\right.$is a continuous function.

Show that the function defined by $f\left(x\right)=\cos \left(x^2\right)$ is a continuous function.

Show that the function defined by $f\left(x\right)=\left|\cos x\right|$ is a continuous function.

Examine that $\sin \left|x\right|$ is a continuous function.

Find all the points of discontinuity of $f$ defined by $f\left(x\right)=\left|x\right|-\left|x+1\right|.$