Mean Value Theorem - f(x) = f(a)+(x-a)f'(u)

Click For Summary

Discussion Overview

The discussion revolves around the Mean Value Theorem, specifically the conditions under which it holds true. Participants explore the implications of continuity and differentiability of a function on the interval [a, b] and the open interval (a, b), as well as the consequences of these conditions on the theorem's applicability.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants question the validity of stating that if a function f is continuous and differentiable on [a, b], then the Mean Value Theorem applies, particularly when x equals a or b.
  • Others argue that continuity on the closed interval [a, b] is necessary to avoid cases where the function may tend to infinity at the endpoints.
  • It is noted that if f is continuous on (a, b), it does not automatically imply continuity on [a, b], as limits at the endpoints must be considered.
  • Some participants assert that differentiability at the endpoints is not required for the theorem to hold, but continuity at those points is essential.
  • One participant provides a counterexample involving a piecewise function to illustrate that continuity does not imply differentiability, emphasizing the need for continuity at the endpoints for the theorem's application.
  • There is a discussion about the definition of continuity and how it relates to the assumptions made in the context of the theorem.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of continuity at the endpoints for the Mean Value Theorem to apply. While some agree on the importance of continuity, others propose that the theorem could still hold under certain conditions, leading to an unresolved debate.

Contextual Notes

Participants highlight that the assumptions regarding continuity and differentiability may depend on the specific function being analyzed, and there are unresolved mathematical implications regarding the limits at the endpoints.

rhia
Messages
37
Reaction score
0
Hi,
If f is continuous in [a,b] and differentiable in (a,b), and if xE(a,b), then there exists u in (a,b) such that
f(x) = f(a)+(x-a)f'(u)


What's wrong if i state :
(i)If f is continuous in [a,b] and differentiable in [a,b], and if xE[a,b]
or
(ii)If f is continuous in (a,b) and differentiable in (a,b), and if xE(a,b)

Thanks!
 
Physics news on Phys.org
usually we write,
f'(u) = (f(x)-f(a))/(x-a) .. (1)

(i)If f is continuous in [a,b] and differentiable in [a,b], and if xE[a,b]
here x can take value of a ..
see what happens to (1) when x takes value of a?

(ii)If f is continuous in (a,b) and differentiable in (a,b), and if xE(a,b)
it is easy to note that if f is continuous in (a,b) then it is continuous in [a,b]
cuz f(a) = lim_{x->a+}f(x)
(we don't bother about x->a-)
similarly,
f(b) = lim_{x->b-}f(x)
(we don't bother about x->b+)

-- AI
 
In the second case, f may tend to infinity at a or at b. So f being continuous on [a, b] is required.
 
TenaliRaman said:
it is easy to note that if f is continuous in (a,b) then it is continuous in [a,b]
cuz f(a) = lim_{x->a+}f(x)
-- AI
f(a)=lim{x->a} f(x) is the definition of continuity at a, so this is just assuming what you are trying to show.

If you don't assume continuity at a then f(a) can have whatever value you like, unrelated to the rest of f(x). e.g.
f(0)=1, f(1)=1,f(x)=0 for x in (0,1).
 
Wong said:
In the second case, f may tend to infinity at a or at b. So f being continuous on [a, b] is required.

in the second case,
f(x) is defined over (a,b) which excludes a and b
so f(a) and f(b) are actually not defined originally and hence cannot be infinity
so i can define continuity extending to both extremities and still the theorem would be true

[Edit]eeps! Wong is right on one thing here
say the function is 1/x and the interval is (0,1) the limx->0+ f(x) would be infinity ... So i suppose we could add a condition that lim x->a+ f(x) and limx->b-f(x) must exist and must be finite... then it is extendible as i said earlier[/Edit]

chronon said:
f(a)=lim{x->a} f(x) is the definition of continuity at a, so this is just assuming what you are trying to show.

No
definition of continuity requires that,
f(a) = limx->a- f(x) = limx->a+ f(x)
(this is a strict necessity)
But if u see what i said in my earlier post, when we talk of an interval like (a,b) i can skip one of the conditions since my viewport now is restricted to interval (a,b) alone.

-- AI
 
Last edited:
so f(a) and f(b) are actually not defined originally and hence cannot be infinity
so i can define continuity extending to both extremities and still the theorem would be true

In other words, you're assuming f is a function continuous on [a, b]. :-p
 
There is nothing WRONG with assuming f is differentiable on [a,b] except that it is not as general. The theorem is true even if f is NOT differentiable at a or b.
(Consider the function f(x)= |x| and apply the mean value theorem on [0, 1].)

On the other hand you MUST have f continuous at a and b since the values used in the theorem are calculated there.

For example, if f(x)= x for x NOT equal to 0 or 1 but f(0)= -1, f(1)= 2. Can you find c between 0 and 1 such that f'(c)= f(1)- f(0) ?
 
Hurkyl said:
In other words, you're assuming f is a function continuous on [a, b]. :-p

Well i did edit my post indicating that fact :biggrin:
Though i am not exactly assuming it :-p
 
HallsofIvy said:
For example, if f(x)= x for x NOT equal to 0 or 1 but f(0)= -1, f(1)= 2. Can you find c between 0 and 1 such that f'(c)= f(1)- f(0) ?

If I got this right, you can't.

This is because it is not continuous at 0 and 1.

Continuity does not imply differentiability.

When they say it is continuous on the interval, they are saying:

f(a)=the limit, as x approaches a, is equal to f(a). Again, it has been mentionned earlier that the right-hand limit must equal the left-hand limit. That is continuity. It may be continuous on the left or right, but the theorem has mentionned nothing about it, so it shall be excluded in this case.

Feel free to write your own theorem.
 

Similar threads

Undergrad Continuity of Mean Value Theorem
  • · Replies 12 ·
Replies
12
Views
3K
Undergrad Uniform convergence and derivatives -- difference between two theorems?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Doubt about theorem in Calculus on Manifolds
  • · Replies 9 ·
Replies
9
Views
4K
Undergrad On inverse function theorem in Spivak's CoM
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Should the function f to be continuous for applying MVTI or not?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Question about proof of Great Picard Theorem
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Proving that convexity implies second order derivative being positive
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Looking for Guarantees that the method of fixed-point iteration will work
  • · Replies 22 ·
Replies
22
Views
3K
Undergrad Showing That a Function Does Not Have Two Distinct Roots
  • · Replies 3 ·
Replies
3
Views
2K