Differentiation and Continuity
In one of our previous pages, we have seen that if f'(x0) exists, then
for x close to x0, we have
This is the "linear approximation" done via the tangent line.
Obviously this implies
which means that f(x) is continuous at x0. Thus there is a
link between continuity and differentiability: If a
function is differentiable at a point, it is also continuous
there. Consequently, there is no need to investigate for
differentiability at a point, if the function fails to be
continuous at that point.
Note that a function may be continuous but not differentiable,
the absolute value function at x0=0 is the archetypical
This relationship between differentiability and continuity is
local. But a global property also holds. Indeed, let f(x) be
a differentiable function on an interval I. Assume that
f'(x) is bounded on I, that is there exists M >0 such that
The Mean Value Theorem will then imply that
This is the definition of Lipschitz
continuity. In other words, if f'(x) is bounded then f(x)is a Lipschitzian function. Conversely, it is also true that
Lipschitzian functions have bounded first derivatives, when they
exist. Since Lipschitzian functions are uniformly continuous,
then f(x) is uniformly continuous provided f'(x) is bounded.
Nevertheless, a function may be uniformly continuous without
having a bounded derivative. For example,
uniformly continuous on [0,1], but its derivative is not bounded
on [0,1], since the function has a vertical tangent at 0.
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