In my textbook on metric spaces by N.L. Carothers, the defintion for a continuous map is as follows (slightly modified by my professor from my lecture notes):
Let $(M,d)$ and $(N,\rho)$ be metric spaces and $f : M \to N$. $f$ is continuous at $x \in M$ $\iff$ $\forall$ $\epsilon >0$ $\exists$ $\delta >0$ (which depends on $f$, $x$, and $\epsilon$) such that $d(x,y) < \delta$ $\implies$ $\rho(f(x),f(y)) < \epsilon$.
This definition is pretty standard, however, there is something that has recently been of question to me as I have gotten further in my studies: what does the author (and my professor) mean when they say $\delta$ depends on $f$? I understand that it depends on your choice of $x$ and $\epsilon$, but this is the only literature I have read where the author says that $\delta$ depends on $f$ as well. I have taken this for granted up until now, but I have been thinking about it a lot.
While uniform continuity is much stronger than continuity, this dependence of $\delta$ on $f$ makes an appearance again in their definitions (while on wikipedia, they state $\delta$ only depends on $\epsilon$):
$f : M \to N$ is uniformly continuous $\iff$ $\forall \epsilon > 0$ $\exists$ $\delta >0$ (which depends on $f$ and $\epsilon$ alone) such that $d(x,y) < \delta$ $\implies$ $\rho(f(x),f(y)) < \epsilon$.
If someone could clear this up for me, it would be much appreciated.
Best Answer
The "usual" definition is
You say it is clear that $\delta$ will (in general) depend on $x$ and $\epsilon$. However, it is also clear that $\delta$ will depend on all parameters occurring in $$d(x,y) < \delta \implies \rho(f(x),f(y)) < \epsilon .$$ These are $x, \epsilon, f, d, \rho$. That is, if you take other metrics $\bar d$ on $M$, $\bar \rho$ on $N$ and another function $\bar f : M \to N$, you cannot expect that the same $\delta$ will do.
Usually the dependency on $f, d, \rho$ is not explicitly mentioned because the first sentence of the definition is "Let $(M,d)$ and $(N,\rho)$ be metric spaces and $f : M \to N$." This fixes these three parameters.
However, I am sure that you aware that this dependency exists. Think about the proof that the sum of two continuous real-valued functions $f_1, f_2 : M \to \mathbb R$ is again continuous. You certainly chose $\delta(x,\epsilon/2,f_1)$ and $\delta(x,\epsilon/2,f_2)$ and define $\delta(x,\epsilon,f_1+f_2) = \min (\delta(x,\epsilon/2,f_1),\delta(x,\epsilon/2,f_1))$.