##
Differentiation and Continuity

In one of our previous pages, we have seen that if *f*'(*x*_{0}) exists, then
for *x* close to *x*_{0}, we have

This is the "linear approximation" done via the tangent line.
Obviously this implies

which means that *f*(*x*) is continuous at *x*_{0}. 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 *x*_{0}=0 is the archetypical
example.

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

for any
.
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,
is
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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