Example 7

Figure 3.4.12

The semicircle

shown in Figure 3.4.12, is continuous on the closed interval [-1, 1]. It is differentiable on the open interval (-1, 1). To see that it is continuous from the right at x = -1, let Δx be positive infinitesimal. Then

Thus

The number inside the radical is positive infinitesimal, so Δy is infinitesimal. This shows that the function is continuous from the right at x = -1. Similar reasoning shows it is continuous from the left at x = 1.