I am inexperienced in real analysis/more formal math, so I am wondering how to address this issue. I have seen certain theorems that are supposed to hold true for any $a > 0$, such as in Markov's Inequality and Epsilon-Delta proofs. I am wondering if this somehow implies that we can actually work with $a = 0$ in these cases. I'm sure there's some discretion needed, but I wanted to see how to address a few examples.
For example, (this relates to some methods I've seen in the epsilon-delta proof formulations), if for every $\varepsilon > 0$ we have that $|f(t) - c| \leq \varepsilon$, does this imply that $f(t) = c$? I could see this if we can take the limit as $\varepsilon \to 0$, but not sure how to show that otherwise. I think the statement is true though (since I've seen it used before) so if this isn't the correct way to show it, I was wondering what else I can do.
Thanks in advance!
Addressing your example: you can argue that $f(t) = c$ without using limits explicitly. Suppose $f(t) \neq c$. Then $|f(t) - c| = \delta > 0$. Since you can also choose $\epsilon = \frac{\delta}{2}$ you obtain $$ \delta = |f(t) - c| \leq \frac{\delta}{2} . $$ which is only possible for $\delta = 0$.
Sometimes you can make similar arguments, if you already know that some property holds for every $a>0$ to obtain the property also for $a=0$.