Difficulty in Mathematical Writing 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!
You need practice in formulating your intutions.  This is a skill that can be learned through practice, but like all skills, you can't begin just by jumping into the deep-end: you need to work consistently at extending your comfort zone, one small step at a time; over time you will then make substantial improvements.  

What form should your practice take?  Well, you have to practice writing careful and complete proofs.   (The word "writing" here is important; it is not enough just to think about it and convince yourself that you could write a correct answer --- rather, sit down with pen and paper and carefully write out the full and correct answer.)

As a practical suggestion of where to begin (in light of your specific problem): rather than beginning directly with a concept like (failure of) uniform continuity, which involves a rather large number of quantifiers, be sure you can completely prove some simpler statements, such as "$1/(x-a)$ is unbounded on $(a,b)$" or "$x^2$ is unbounded on $\mathbb R$".   Once you can comfortably prove such statements, you can build up to carefully proving analytic statements with more quantifiers.

One more thing: When trying to prove statements (using precise defintions)
which you feel that you have a good intuitive understanding for, try to self-conciously refelect on how your intuition accords with the formal definitions.  For example, ask yourself "why do I think that $x^2$ is unbounded on $\mathbb R$?" Try to analyze your own intuition and extract some statement out of it.  Then compare this with the formal defintion of unbounded and see if they are the same.  If so, great (!) ; you've got good intuition.  If they dont' quite agree, try to understand why; what was missing in your intuition?  what extra aspect of the situation is the formal definition trying to capture?  Again, reflecting on and analyzing your mathematical intuition is a skill that can be practiced and which you can improve --- but, like all skills, you have to practice it if you want to get better.
The problem isn't really about writing. It's about being able to do math.

One of the big problems in math is the P = NP problem. Most mathematicians believe that the know that the two aren't equal.
There's however no accepted proof. As a result a lot of smart people work on the problem. Writing proofs is the main activity of professional mathematicians.

If you 'or'-sign instead of the 'and'-sign because you don't remember with is turned with way, then you have a problem with writing stuff down in a formal matter. 
If you however have no idea about what theorems you need to use to prove the solution then you misstate your problem by saying it's about writing stuff down in a formal matter.

Math centers around finding a way to prove your result. Faith isn't enough.