Discontinuous Curve

The functions we shall ordinarily encounter in this book will be defined and have a derivative at all but perhaps a finite number of points of an interval. The graph of such a function will be a smooth curve where the derivative exists. At points where the curve has a sharp corner (like 0 in |x|) or a vertical tangent line (like 0 in x^1/3), the function is continuous but not differentiable (see Figure 3.4.7). At points where the function is undefined or there is a jump, or the value approaches infinity or oscillates wildly, the function is discontinuous (see Figure 3.4.8).

Figure 3.4.8: Points where f is discontinuous