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Theorem 3:
Our next theorem is that differentiability implies continuity. That is, the set of differentiable functions at c is a subset of the set of continuous functions at c. (See Figure 3.4.3.)
Figure 3.4.3 THEOREM 3 If f is differentiable at c then f is continuous at c. PROOF Let y = f(x), and let Δx be a nonzero infinitesimal. Then Δy/Δx is infinitely close to f'(c) and is therefore finite. Thus Δy = Δx(Δy/Δx) is the product of an infinitesimal and a finite number, so Δy is infinitesimal. For example, the transcendental functions sin x, cos x, ex are continuous for all x, and ln x is continuous for x > 0. Theorem 3 can be used to show that combinations of these functions are continuous.
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Last Update: 2010-11-25