Question

Show that f, given by $f\left(x\right)=\frac{x-|x|}{x}(x\ne 0)$, is continuous on R$\left[0\right]$.

Answer 1

$$\begin{array}{c}f\left(x\right)=\frac{x-|x|}{x}\\ check\\ f\left(0^{+}\right)\to f_{limh\to 0}\left(0+h\right)\\ R.H.L\\ f\left(0^{+}\right)=f_{leth=0}\left(0+h\right)=\frac{\left(0+h\right)-\left|0+h\right|}{\left(0+h\right)}\\ {=}_{limh\to 0}\frac{l-h}{h}=0\\ L.H.L.\\ f\left(0^{-}\right)=f_{leth=0}\left(0-h\right)=\frac{\left(0-h\right)-\left|0-h\right|}{\left(0-h\right)}\\ =\frac{-h-h}{-h}\\ =\frac{-\left(2h\right)}{-h}\\ =2\end{array}$$