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I am trying to answer the following question from my book:

Propose a definition for what it should mean for $f: A \to \mathbb R$ to be uniformly differentiable on $A$.

Since the definition of uniform continuity is the following:

A function $f: A \to \mathbb R$ is uniformly continuous on $A$ if and only if for $\varepsilon > 0$ there exists $\delta$ such that $|x-y|< \delta$ implies $|f(x) -f(y)|<\varepsilon$

I was thinking of the following:

A (differentiable) function $f: A \to \mathbb R$ is uniformly differentiable on $A$ if and only if for $\varepsilon > 0$ there exists $\delta$ such that $|x-y|<\delta$ implies $|f'(x) -f'(y)|<\varepsilon$.

But this is just saying $f'$ is uniformly continuous. So the next thing I came up with is:

A (differentiable) function $f: A \to \mathbb R$ is uniformly differentiable on $A$ if and only if there exists $\varepsilon > 0$ such that for all $x\in A$ it holds that $|f'(x)|<\varepsilon$.

But this is just saying the derivative is bounded. Now I'm stuck. How do I go about finding a suitable definition?

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  • $\begingroup$ Perhaps, give some condition on how well the difference quotients approximate the value of the derivative. $\endgroup$ – David Mitra Feb 8 '14 at 12:16
  • $\begingroup$ It is worth remembering that Cauchy's proof of the MVT (1823) contains an invalid use of this concept. $\endgroup$ – Tony Piccolo Nov 22 '15 at 16:49
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The most reasonable definition:

Let $A\subset \mathbb R$ and $f: A\to \mathbb R$. We say that $f$ is uniformly differentiable on $A$, is for every $\varepsilon>0$, there exists a $\delta>0$ such that, for every $x\in A$, $$ |h|<\delta\,\,\text{and}\,\,x+h\in A\quad\Longrightarrow\quad \left|\frac{f(x+h)-f(x)}{h}-f'(x)\,\right|<\varepsilon. $$

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This is such a common notion in analysis that I think it is worth elaborating on, at least when I taught analysis at this level I would do so.

The first starting point is historical. Cauchy was the first to define continuity as we now know it although the word was used loosely before him. Here is his definition (that you already well know):

A function $f:[a,b]\to\mathbb{R}$ is continuous, if for each point $x$ in $[a,b]$ and for any $\epsilon>0$ there is a $\delta>0$ so that $|x-y|<\delta $ $\implies$ $|f(x)-f(y)|<\epsilon$.

Then, when he set about to use this concept he switched without noticing this to a different version:

A function $f:[a,b]\to\mathbb{R}$ is [uniformly] continuous, if for any $\epsilon>0$ there is a $\delta>0$ so that $|x-y|<\delta $ $\implies$ $|f(x)-f(y)|<\epsilon$ for each point $x$ in $[a,b]$.

All that happens, of course, is that the quantifier "for each point" moves from the beginning of the statement to the end, with dramatic consequences. The first definition we now call "pointwise continuity" and the second (much stronger) definition we now call "uniform continuity." Actually Cauchy's mistake was quite fruitful. He needed the stronger condition but ended up giving us two useful definitions (although he thought they were the same).

Here is another example which sometimes helps the intuition.

Let $A$ and $B$ be the collection of all math students at two universities. We say that University B is "smarter" than University A if for each student $s_A$ in $A$ there is a subject and a student $s_B$ in $B$ so that $s_B$ has higher grades than $s_A$ in that subject.

Let $A$ and $B$ be the collection of all math students at two universities. We say that University B is "uniformly smarter" than University A if there is a subject and a student $s_B$ in $B$ so that $s_B$ has higher grades than $s_A$ in that subject for each student $s_A$ in $A$.

Again all that is happening is that the quantifier (for each student $s_A$) has moved to the end of the statement, but the difference is again dramatic.

The exercise here (left to the reader) is to explain in words what the difference is between the uniform version and the "pointwise" version. Think, too, about what would you have to do to prove or disprove the following assertions:

Yale Math is smarter than Harvard Math.

Yale Math is uniformly smarter that Harvard Math.

When you look at the other answer to this question provided by Mr Smyrlis, make sure to see that he has reproduced the completely familiar definition of a derivative happening at each point $x$ and then moved the quantifier to the end of the definition thus obtaining "uniform differentiability." [In fact he just made sure to move it to after the choice of $\delta$.]

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  • $\begingroup$ If you are somewhat braver and don't mind being provocative try this version: $U$ is the set of students at a university and there is a relation $s\heartsuit t$ defined for some pairs $s$, $t$ in $U$. [Not reflexive, probably not symmetric, definitely not transitive.] We say $U$ is a happy university if for each $s\in U$ there exists a $t\in U$ so that $s\heartsuit t$. Define what would be meant by a uniformly happy university and explain in words the difference between the two concepts. $\endgroup$ – B. S. Thomson Nov 21 '15 at 18:30

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