Real Analysis: Uniformly continuous

In summary, the conversation discusses the continuity and integrability of a real valued function on the set of real numbers, R, and its corresponding integral function, F. It is mentioned that F is uniformly continuous on R and this can be proven using the Second Fundamental Theorem of Calculus and the fact that F has a bounded derivative. The speaker also suggests considering a different approach to understand uniform continuity.
  • #1
annastm
4
0
Please help me with this one:

Suppose f is a continuous real valued function on R - real #s and that 0<f(x)<3 for every x in R. Let F(x) = integral from 0 to x of f(t) dt.

a) Show that F is uniformly continuous on R
b) Suppose the continuity assumption is left out, but the function f is still assumed to have 0<f(x)<3 for all x and to be integrable on every bounded interval. Must the conclusion of uniform continuity of F on R still be true? Prove or give counterexample.

Thank You for your help
 
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  • #2
By teh (second?) Fundamental Theorem of Calculus, F, in part a), is a differentiable function with (uniformly) bounded derivative. It is a standard exercise to show this is uniformly continuous, so what do you think you need to do now you've got this (slightly) different way of thinking about it?
 

What is Real Analysis?

Real Analysis is a branch of mathematics that deals with the study of real numbers and their properties. It involves the use of mathematical proofs and techniques to understand the behavior and characteristics of real numbers and functions.

What is Uniform Continuity?

Uniform Continuity is a concept in real analysis that describes the behavior of a function as the input values get closer to each other. A function is considered uniformly continuous if, for any two points in its domain, the difference between the output values can be made arbitrarily small by making the input values close enough to each other.

How is Uniform Continuity different from Continuity?

Uniform Continuity and Continuity are both concepts that describe how a function behaves near a point. However, uniform continuity is a stronger condition than continuity. While continuity only requires the function to have no sudden jumps or breaks, uniform continuity also requires that the function's behavior is consistent no matter how close the input values are.

What is the importance of Uniform Continuity in Real Analysis?

Uniform Continuity is an essential concept in real analysis because it allows us to make precise statements about the behavior of functions. It allows us to prove theorems and make deductions about the properties of functions that are uniformly continuous. Without this concept, it would be challenging to analyze the behavior of functions and understand their properties.

How is Uniform Continuity related to Lipschitz Continuity?

Lipschitz Continuity is a stronger form of uniform continuity. A function is considered Lipschitz Continuous if there is a finite number that bounds the ratio between the difference in output values and the difference in input values. This means that a Lipschitz Continuous function is also uniformly continuous, but the reverse is not always true.

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