Visualizing (In-)Continuity

When f is continuous at c, the entire part of the curve where x ≈ c will be visible in an infinitesimal microscope aimed at the point (c, f(c)), as shown in Figure 3.4.1(a). But if f is discontinuous at c, some values of f(x) where x ≈ c will either be undefined or outside the range of vision of the microscope, as in Figure 3.4.1(b).

Figure 3.4.1

Continuity, like the derivative, can be expressed in terms of limits. Again the proof is immediate from the definitions.