I am struggling to understand a basic definition of a continuous function from a textbook:
A function f is continuous if for all x, and for all $\epsilon>0$, there exists $\delta>0$ such that for all y, if $|x-y|<\delta$, then $|f(x)-f(y)|<\epsilon$
In particular, the "for all y" (hasn't choosing an $\epsilon$ constrained this?) and the use of implication does not seem to fit with my understanding of a continuous function, which is something like:
For all x and $\epsilon>0$ (esp. arbitrarily small; so I agree so far), you can always satisfy the inequality $|f(x)-f(y)|<\epsilon$ with a small enough $\delta$ (i.e. constraining y as in $|x-y|<\delta$ above). If this non-technical definition is actually correct, how can I relate to that above?
The definition you give is correct.
In the definition above it says "for all $y$, if $|x-y|<\delta$ then ...". That actually means: for all $y$ with the constraing being close enough to $x$, as you describe, the following part will be true (in your case: $|f(x)-f(y)|<\delta$.