INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

September 12, 2013

Differentiability and Uniform Continuity

Problem: Every differentiable function f:  (0, 1) --> [0, 1] is uniformly continuous.



Note that every differentiable function f: [0,1] --> (0, 1) is uniformly continuous by virtue of uniform continuity theorem which says every continuous map from closed bounded interval to R is uniformly continuous. However in this case the domain is an open interval.

We can easily find counter example such as f(x) = \sin ( \frac {1}{x} ) . Intuitively speaking the function oscillates (between -1 and 1) faster and faster as we get close to x = 0. Hence we can get two arbitrarily close values of x such that their functional value's difference equals a particular number (say 1) therefore exceeding any \epsilon < 1

An interesting discussion:

differentiability and uniform continuity

One comment on “Differentiability and Uniform Continuity”

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.