Real Analysis question - Show that the derivative is continuous.

In summary, the problem states that if a function f is differentiable at every point in a closed, bounded interval and its derivative is increasing on that interval, then the derivative must also be continuous on that interval. This can be proven by using the Intermediate Value Theorem for Derivatives and showing that a discontinuity in the derivative would contradict with the theorem.
  • #1
glacier302
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Homework Statement



Suppose that f is differentiable at every point in a closed, bounded interval [a,b]. Prove that if f' is increasing on (a,b), then f' is continuous on (a,b).

Homework Equations



If f' is increasing on (a,b) and c belongs to (a,b), then f'(c+) and f'(c-) exist, and f'(c-) <= f'(c) <= f'(c+).

IVT for Derivatives (also called Darboux's Theorem): Suppose that f is differentiable on [a,b] with f'(a) not equal to f'(b). If y0 is a real number which lies between f'(a) and f'(b), then there is an x0 belonging to (a,b) such that f'(x0) = y0.

The Attempt at a Solution



So I know that since f' is increasing, for any c in (a,b), f'(c-) <= f'(c) <= f'(c+). So for any h > 0, f'(c-h) <= f'(c) <= f'(c+h). If f'(c-h) is not equal to f'(c+h), then since f'(c) lies between f'(c-h) and f'(c+h), by the Intermediate Value Theorem for Derivatives there is an x0 belonging to (c-h, c+h) such that f'(x0) = f'(c).

My thought is that if I can show that the only x0 in (c-h,c+h) such that f'(x0) = f'(c) is c, then the limit of f'(x) as x approaches c must be f'(c), which means that f' is continuous at c. But how do I show that the only x0 in (c-h,c+h) such that f'(x0) = f'(c) is c?

Thank you in advance for any help!
 
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  • #2
I think you are on the right track. You probably can show that a increasing f' with a discountinuity would contradict with Darboux theorem.
 

1. What is "Real Analysis"?

Real Analysis is a branch of mathematics that deals with the study of real numbers and their properties. It is concerned with the rigorous development and understanding of concepts such as limits, continuity, derivatives, and integrals.

2. What is a derivative?

A derivative is a mathematical concept that represents the instantaneous rate of change of a function at a specific point. It is defined as the slope of the tangent line to the function at that point and is used to calculate the rate of change of a function at any given point.

3. How do you show that a derivative is continuous?

To show that a derivative is continuous, we need to prove that it satisfies the definition of continuity. This means that the limit of the derivative as the point approaches a given value must be equal to the derivative at that point. In other words, the derivative must not have any sudden jumps or breaks in its graph.

4. What is the importance of a continuous derivative?

A continuous derivative is important because it ensures that the function is smooth and well-behaved. It also allows us to make accurate predictions about the behavior of the function at various points and to calculate important quantities such as rates of change and maximum and minimum values.

5. Can you give an example of a function with a continuous derivative?

An example of a function with a continuous derivative is the simple polynomial function f(x) = x^2. Its derivative, f'(x) = 2x, is continuous at all points on the real number line. This means that the function is differentiable at all points and its graph is smooth and without any sudden jumps or breaks.

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