I was approaching the following problem:
"Let $f \colon X \to Y$ be continuous. Is it true that if $x$ is a limit point of $A \subset X$ then $f(x)$ is a limit point of $f(A)$?"
The answer is that it is false and here is a counterexample I found: $X = \mathbb{R}$ with the standard topology, $Y = \mathbb{N}$ with the discrete topology and finally $f(x) = 1$ for every $x \in \mathbb{R}$. (it's a counter example since $2$ is a limit point of $[0,58]$ but $f(2)$ is not a limit point of $f([0,58])$. To prove this just notice that $\{1\}$ is an open neighborhood of $f(2)$ but $\{1\}\cap (f([0,58]) \setminus \{f(2)\}) = \emptyset$.
Now you are all thinking "where's the problem then?"... The problem is that my intuition failed and I tried to prove the statement for a while before trying to find a counterexample!
What can I do to avoid this problem in future? Is there any intuition I should have had to start looking for counterexamples before trying to prove the affirmative result?