Question

Explain the following steps to show that a function that is continuous at every point of a closed interval is uniformly continuous on that interval. It will be a proof by contradiction, so we assume that f is continuous, but not uniformly continuous, on [a,b]. (a) There must be some ε>0 and for each n=1,2,3,…, two numbers xn ,yn in [a,b] for which ∣xn−yn∣< n1 and ∣f(xn )−f(yn)∣≥ε.

Answer 1

To prove that a function that is continuous at every point of a closed interval is uniformly continuous on that interval, we assume that f is continuous but not uniformly continuous on [a,b]. By using the contrapositive, we can show that if a function is continuous at every point on a closed interval [a,b], then it must be uniformly continuous on that interval.

Step-by-step explanation:

To prove that a function that is continuous at every point of a closed interval is uniformly continuous on that interval, we assume that f is continuous but not uniformly continuous on [a,b].

Next, we assume that there exists some ε>0, and for each n=1,2,3,..., there exist two numbers xn and yn in [a,b] such that |xn-yn|

By using the contrapositive, we can show that if a function is continuous at every point on a closed interval [a,b], then it must be uniformly continuous on that interval.