Question

Construct the graph of the function given below:$f\left(x\right)=\{\begin{array}{cc}x-1,& x<0\\ \frac{1}{4},& x=0\\ x^2,& x>0\end{array}$ Find $f\left(x\right)$ and ${lim}_{x\to 0-}f\left(x\right).$

Answer 1

The graph consists of the straight line $y=x-1$ for $x<0$, the point $(0,1/4)$ and the parabola $y=x^{2}forx>0.$
Now $\underset{x\to 0+}{lim}f\left(x\right)=\underset{h\to 0}{lim}{(0+h)}^2=0$ and $\underset{x\to 0-}{lim}f\left(x\right)=\underset{h\to 0}{lim}(0-h-1)=-1.$
Since $\underset{x\to 0+}{lim}f\left(x\right)\ne \underset{h\to 0-}{lim}f\left(x\right),$
the function $f\left(x\right)$ is discontinuous at$x=0.$