Differentiability And Continuity

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Differentiability And Continuity

In this section we shall derive an important relationship between differentiability and continuity, namely.

If f is differentiable at a point, then f is continuous at that point.

To establish this result, suppose that f is differentiable at x = c Then f ’ (c) exists and

f ' (c) = lim  f (c+ h)-f (c)
         h→0       h


lim [f (c+ h)-f (c)] = lim [ f (c+ h)- f(c) . h ]
                          h→0                h

                       = lim f (c+ h)- f(c)  . lim h = f '(c) . 0 =0

Thus lim [f (c+h)-f (c)] = 0. This means that f (c+h)-f (c) approaches 0 as h ( ) 0. Consequently, lim f (c+h) = f (c), i.e., lim f (x) = f (c). This proves that f is continuous at x = c.
                      x →c

Note. The converse of the above result is not necessarily true. That is, a function may be continuous at a point without being differentiable at  that point. For example, the function f (x) = |x| is continuous at x = 0 but not differentiable at x = 0.

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