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Let f:R->R satisfy |f(t)-f(x)|<=|t-x|^2 for any t,x.
Prove that (f) is constant.

© BrainMass Inc. brainmass.com September 29, 2022, 2:29 pm ad1c9bdddf
https://brainmass.com/math/derivatives/real-functions-real-variable-continuity-152302

## SOLUTION This solution is FREE courtesy of BrainMass!

We don't know anything about function (f), whether it is or it is not continuous, differentiable etc.
We just know that the above inequality is valid for any values of (t) and (x).
Let's consider the special case when
(1)
&#61664;
(&#61541; = small quantity) (2)
From (1) and (2), we conclude that f = uniform continuous.
Let's write now the inequality from our problem as below:
(3)
If we apply the condition (1), then we will have:
(4)
Therefore, we can put
(5)
so that (4) becomes
(6)
Since f(x) = continuous, we take the limit when &#61541; &#61614; 0 and one can see that the term of the left side defines the derivative of f(x), which in this case exists and is null:
(7)
Since (as a modulus) and x is any real number, from the above equation we conclude that

&#61664;

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

© BrainMass Inc. brainmass.com September 29, 2022, 2:29 pm ad1c9bdddf>
https://brainmass.com/math/derivatives/real-functions-real-variable-continuity-152302