Real Analysis: Uniformly continuous

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SUMMARY

The discussion centers on the uniform continuity of the function F(x), defined as the integral of a continuous real-valued function f(x) that satisfies 0 < f(x) < 3 for all real numbers x. It is established that F is uniformly continuous on R due to the bounded derivative derived from the Fundamental Theorem of Calculus. Furthermore, the discussion raises the question of whether F remains uniformly continuous if the continuity of f is relaxed, while still maintaining the condition 0 < f(x) < 3 and integrability over bounded intervals, prompting the need for a counterexample or proof.

PREREQUISITES
  • Understanding of real-valued functions and their properties
  • Familiarity with the Fundamental Theorem of Calculus
  • Knowledge of uniform continuity and its implications
  • Concept of integrability over bounded intervals
NEXT STEPS
  • Study the properties of uniformly continuous functions in depth
  • Explore counterexamples related to uniform continuity without continuity assumptions
  • Review the implications of the Fundamental Theorem of Calculus on function continuity
  • Investigate the conditions under which integrable functions are uniformly continuous
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Mathematicians, students of real analysis, and educators looking to deepen their understanding of uniform continuity and its relationship with integrable functions.

annastm
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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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By the (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?
 

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