Prove that if $f(x)$ is continuous on the closed interval $[a,b]$, then $f(x)$ is uniformly continuous on $[a,b]$.

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Lesson Parent: 
Uniform Continuity
Problem: 

Prove that if $f(x)$ is continuous on the closed interval $[a,b]$, then $f(x)$ is uniformly continuous on $[a,b]$.

Answer: 

It is true that if $f(x)$ is continuous on the closed interval $[a,b]$, then $f(x)$ is uniformly continuous on $[a,b]$.

SAMPLE SOLUTION

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Solution: 

This proof is extremely hard and almost never required for a beginning Calculus course - even a beginning Calculus course for mathematics majors. You can safely ignore this proof, but be aware that it is an important proof for showing that continuous functions are integrable.

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