Determine whether each of the following functions is uniformly continuous on the given interval.

Question

I am totally lost on this question. I have looked at similar examples and just do not understand how to solve the following examples. Thank you in advance for any help!

a.) $f(x) = x^4-4x+7, I= [1,2]$

b.) $f(x) = \frac {x+2}x, I= (0,1]$

Answer 1

There is a theorem that states that every continuous function defined in a compact set is uniformly continuous. The difference is that on uniform continuity, $\delta$ does not depend on $x$, it only depends on $\varepsilon$. Given a function $f$, we say that $k \in \mathbb{R}$ is a Lipschitz constant for $f$ if $|f(x) - f(y)| \leq k |x - y|$, for all $x,y$ in $f$'s domain. Every Lipschitz function is uniformly continuous. (the converse is false!) If the function is defined on an interval, and has limited derivative, let's say, $|f'(x)| \leq k$, for all $x$ in the interval, then we have $$|f(x) - f(y)| = |f'(c) (x - y)| = |f'(c)||x-y| \leq k|x - y|$$ for some $c$ between $x$ and $y$. Try to use all this information. Ok?