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If we want to indicate the value of a derivative
dy
dx
in Leibniz
notation at a specific number a, we use the notation
dy
dx
x=a
or
dy
dx
x=a
which is a synonym for f
(a).
Definition 17
A function f(x) is differentiable at a if f
(a) exists. It is differentiable
on an open interval (a, b) [or (a, +∞) or (−∞,a) or (−∞, +∞)].
if it is differentiable at every number in the interval.
Both continuity and differentiability are desirable properties for a
function to have. The following theorem shows how these properties
are related.
Theorem 7
If f is differentiable at a,thenf is continuous at a.
Proof
Clearly, the difference f(x)−f(x
0
) tends to zero (where x tends
to x
0
) since the limit of the difference quotient exists. So, f(x) tends
to f(x
0
) as x goes to x
0
. This means continuity of f at x
0
.
How can a function fail to be differentiable? The above theorem
implies that if f is not continuous at a, then f is not differentiable at
a. So at any discontinuity f fails to be differentiable. In the examples
(below) we discuss two situations where a function is continuous at
a given number but it fails to be differentiable there.
Example 10
Consider the following function:
f(x)=
x, if 0 x 1,
2x − 1, if 1 <x 2.
Let us show that y = f(x) is continuous at 1, but it is not differentiable
there.
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