Determining Where a Function Is Continuous and Differentiable from a Graph

The graph of $f\left(x\right)$ is continuous on $\left(\mathrm{-\infty ,}\mathrm{-2}\right)\cup \left(\mathrm{-2}\mathrm{,\; 1}\right)\cup (\mathrm{1,}\infty ).$ The graph of $f\left(x\right)$ has a removable discontinuity at $x=\mathrm{-2}$ and a jump discontinuity at $x=1.$ See Figure 16.

Figure 16 Three intervals where the function is continuous

The graph of is differentiable on $\left(\mathrm{-\infty ,}\mathrm{-2}\right)\cup \left(\mathrm{-2,}\mathrm{-1}\right)\cup \left(\mathrm{-1,}1\right)\cup \left(\mathrm{1,}<mn>2</mn>\right)\cup \left(\mathrm{2,}\infty \right).$ The graph of $f(x)$ is not differentiable at $x=\mathrm{-2}$ because it is a point of discontinuity, at $x=\mathrm{-1}$ because of a sharp corner, at $x=1$ because it is a point of discontinuity, and at $x=2$ because of a sharp corner. See Figure 17.

Figure 17 Five intervals where the function is differentiable