TITLE: Difficulty in Mathematical Writing
QUESTION [17 upvotes]: Lots of people (including myself) face lot of problems in tackling Mathematical Problems, which appear as if we can solve it, but then writing out a solution becomes difficult.
Let us consider some examples:

I was asked this question, some time back in an exam. Give an example of a continuous function on $(a,b)$ which is not uniformly continuous. Well, one's obvious choice is $$f(x) = \frac{1}{x-a} \quad \text{or} \ \frac{1}{x-b}$$ I knew this as soon as I saw the problem, and started proving it. One actually has to make an observation that as $x \to a$ then, $\frac{1}{x-a}$ will be larger. But I found that I couldn't actually formally prove it.

Similarly, to prove that $f(x)=x^{2}$ is not uniformly continuous on $\mathbb{R}$, one again has to play with the quantifiers, to get the contradiction part.


So, these are two instances, where I have found the problem, which to me appeared that I could solve it, but writing out a formal solution became difficult.
How can students improve upon this? Are there any instances, which happened to you like this!

REPLY [2 votes]: I know this may be a little late, but I'm in the middle of an extremely good book, "How To Prove It - A Structured Approach".  I'm just starting a compSci degree and I've found the book very practically helpful in understanding what it takes to write a good proof and what strategies to use when approaching writing a proof.