The limit as a function on functions?

Question

Is the observation that, "The limit is a function on functions," true or meaningful? Since the limit is unique, if it exists, then the pair $\displaystyle(c,\lim_{x\to c}f(x))$ itself defines a function for some function $f$. I was just thinking about the idea and wanted some outside feedback.

Answer 1

This is true (in a sense) but probably not often useful. If $f$ is continuous the function whose graph is $(c, \lim_{x \to c} f(x))$ is just $f$.

There is, however, a related idea that's very important. If you have a set of functions with the same domain and codomain then for each $c$ you can consider the evaluation map whose graph is the set $$ (f, f(c)) . $$ That function of functions has many uses and far reaching generalizations.