Question

Find if the given below function is continuous or discontinuous at the indicated point
$f\left(x\right)=|x|+|x-1|atx=1$

Answer 1

Use the concept of continuity of a function at a point
$f\left(x\right)=|x|+|x-1|\cdots \left(1\right)$
Find LHL, RHL and value of function (if required) at given point and compare them for checking continuity:
At $x=1$ $L.H.L={lim}_{x\to 1-}f\left(x\right)={lim}_{h\to 0+}f(1-h)$
${lim}_{h\to 0+}[|1-h|+|1-h-1|]$ $=1+0=\mathrm{1....}\left(2\right)$
$R.H.L={lim}_{x\to 1+}f\left(x\right)={lim}_{h\to 0+}f(1+h)$
$={lim}_{h\to 0+}[|1+h|+|1+h-1|]$
$=1+0=\mathrm{1..}\left(3\right)$
Also,$f\left(x\right)=|1|+|0|=\mathrm{1...}\left(4\right)$
From$\left(2\right),\left(3\right),\left(4\right)$
$L.H.L=R.H.L=f\left(1\right)$
Hence,$f\left(x\right)$ is continuous at $x=1$