Eventually when studying mathematics, you will encounter moments that require rigor. This means your statements (claims, to be loose) should be formulated in a way that it is understood cohesively and consistently to any random third mathematician.

The intuition that motivates your claim is always an important one, but the final claim (or lemma) must be stated precisely without any potential confusion from the reader.

For example, if your intuition says: ‘A smooth curve is always connected’, you should be more rigorous on what you mean by ‘smooth’, a ‘curve’, and ‘connected’. This is because each term can be interpreted differently in various scopes of mathematics.

One may reduce confusion be stating: A curve

$$ \gamma:[a,b]\rightarrow M $$

everywhere infinitely differentiable is globally continuous on (a,b).

You must also describe what M is. Is it a manifold? A Euclidean Space? (If you want to be even more rigorous, you can specify the topology on M).

Such disambiguation is vital not only for the third party but also to yourself. Often what comes after a claim is a proof, but with a poorly stated lemma it’s even impossible for the poser to prove it- as he or she themselves are uncertain of ‘what’ to prove.

Such process does not only restrict to creating theorems. We can go much far by rigorously reformulating a given math problem. This manifests the target we must try to tackle down.

For instance, if the problem says : ‘the function has two horizontal intercepts with a single vertical asymptote’, we may reformulate it rigorously into two conditions,

$$ f(x)=0 $$

Has two distinct roots and there is a unique

$$ x=x_s $$