Question

Find if the given function is continuous or discontinuous at the indicated point
$f\left(x\right)=\{\begin{array}{cc}|x|\cos \frac{1}{x},\text{if}x\ne 0& \\ 0,\text{if}x=0\text{at}x=0& \end{array}$

Answer 1

Use the concept of continuity of a function at a point Given that $f\left(x\right)=\{\begin{array}{cc}|x|\cos \frac{1}{x},\text{if}x\ne 0& \\ 0,\text{if}x=0\text{at}x=0& \end{array}$
Find LHL, RHL and value of function (if required) at given point and compare them for checking continuity:
At $x=0$
$\text{LHL}=\underset{x\to 0-}{lim}f\left(x\right)=\underset{h\to 0+}{lim}f(0-h)$
$=\underset{h\to 0+}{lim}|0-h|\cos \frac{1}{(0-h)}$
$=\underset{h\to 0+}{lim}h\cos \frac{1}{h}$
$\cos (-\theta )=\cos \theta and|-h|=-(-h)=hash>o$
$=\mathrm{0....}\left(2\right)$
{range of cosine function is $-1$ to $1$}
$R.H.L={lim}_{x\to 0+}f\left(x\right)={lim}_{h\to 0+}f(0+h)$
$={lim}_{h\to 0+}|0+h|\cos \frac{1}{0+h}$
$={lim}_{h\to 0+}h\cos \frac{1}{h}$
$=\mathrm{0....}\left(3\right)$
{range of cosine function is $-1\text{to}1$}
Also $f\left(0\right)=\mathrm{0.....}\left(4\right)$
Using $\left(2\right),\left(3\right),\left(4\right)$
$L.HL=R.H.L=f\left(0\right)$
Hence,$f\left(x\right)$ is continuous at $x=0$