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** Up:** Functions, continuity and differentiability
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We show the important properties called the intermediate value
property and extremal value property of continuous functions. We also
deduce the clutch of theorems called mean value theorem, Rolle's
theorem and so on for differentiable functions.
The following important property of continuous functions will be used
all the time.

**Exercise 39**
Let *f* (*x*) be a continuous function and {*x*_{n}} be a sequence
converging to *c*, then {*f* (*x*_{n}} is a sequence converging to
*f* (*c*). (Hint: Examine the condition for continuity near *c*).

Let *f* (*x*) be continuous for *x* satisfying
*a* *x* *b*. Let *c*
be a real number lying between *f* (*a*) and *f* (*b*) we want to show that
*c* is a value of *f*; in other words any number intermediate to two
values is itself a value.

**Exercise 40**
Let

*s* be the least upper bound of the set

Show that

*f* (

*s*) =

*c*.(Hint: To show that

*f* (

*s*)

*c* take a sequence
of points approaching

*s* from above).

Now let *C* be the least upper bound of the values of *f* (*x*), i. e. it is the least upper bound of the set
{*f* (*x*)| *a* *x* *b*}. The
*C* is an extremal value for *f*.

**Exercise 41**
Show that

*C* =

*f* (

*x*) for some

*x* in the range

*a* *x* *b*. (Hint: We have a sequence {

*x*_{n}} so that

*f* (

*x*_{n}) converges
to

*C*; by the section on sequences this sequence has a convergent
subsequence).

The following property of differentiable functions is very important

**Exercise 42**
Let

*f* (

*x*) be differentiable for

*x* satisfying

*a* *x* *b*. Let

*s* be such that

*f* (

*s*) is an extremal value for

*f*. Then

*f'*(

*s*) = 0. (Hint: Examine the condition for differentiability near

*s*).

Now suppose that *f* (*x*) is differentiable in the range *a* < *x* < *b* and
continuous at the endpoints *a* and *b* as well. Suppose that
*f* (*a*) = *f* (*b*) = 0. There is a point *s* where *f* attains its maximal
value; similarly there is a point *t* where *f* attains its minimal
value. If *s* is the point *a* or *b* then
*f* (*x*) 0 and if *t* is
the point *a* or *b* then
*f* (*x*) 0. Thus in case both of these
occur then *f* (*x*) = 0 for all *x*; then let *c* = (*a* + *b*)/2. Otherwise let
*c* be any one of *s* and *t* which is not *a* or *b*. Thus we have a
point where *f'*(*c*) = 0.

**Exercise 43**
For a general function

*g*(

*x*) which is differentiable in the range

*a* <

*x* <

*b* and continuous at the endpoints we apply this to the
function

*f* (

*x*) =

*g*(

*x*) +

^{ . }(

*x* -

*a*)

to show that there is a point

*c* where

*g'*(

*c*) =

** Next:** New Functions from old
** Up:** Functions, continuity and differentiability
** Previous:** Definitions
Kapil H. Paranjape
2001-01-20