linguistic:paraphrasing
Continuous functions

An important class of functions when considering limits are continuous functions. These are precisely those functions which preserve limits, in the sense that if f {\displaystyle f} is a continuous function, then whenever a n → a {\displaystyle a_{n}\rightarrow a} in the domain of f {\displaystyle f}, then the limit f (a n) {\displaystyle f(a_{n})} exists and furthermore is f (a) {\displaystyle f(a)}.

In the most general setting of topological spaces, a short proof is given below:

Let f: X → Y {\displaystyle f:X\rightarrow Y} be a continuous function between topological spaces X {\displaystyle X} and Y {\displaystyle Y}. By definition, for each open set V {\displaystyle V} in Y {\displaystyle Y}, the preimage f − 1 (V) {\displaystyle f^{-1}(V)} is open in X {\displaystyle X}.

Now suppose a n → a {\displaystyle a_{n}\rightarrow a} is a sequence with limit a {\displaystyle a} in X {\displaystyle X}. Then f (a n) {\displaystyle f(a_{n})} is a sequence in Y {\displaystyle Y}, and f (a) {\displaystyle f(a)} is some point.

Choose a neighborhood V {\displaystyle V} of f (a) {\displaystyle f(a)}. Then f − 1 (V) {\displaystyle f^{-1}(V)} is an open set (by continuity of f {\displaystyle f}) which in particular contains a {\displaystyle a}, and therefore f − 1 (V) {\displaystyle f^{-1}(V)} is a neighborhood of a {\displaystyle a}. By the convergence of a n {\displaystyle a_{n}} to a {\displaystyle a}, there exists an N {\displaystyle N} such that for n > N {\displaystyle n>N}, we have a n ∈ f − 1 (V) {\displaystyle a_{n}\in f^{-1}(V)}.

Then applying f {\displaystyle f} to both sides gives that, for the same N {\displaystyle N}, for each n > N {\displaystyle n>N} we have f (a n) ∈ V {\displaystyle f(a_{n})\in V}. Originally V {\displaystyle V} was an arbitrary neighborhood of f (a) {\displaystyle f(a)}, so f (a n) → f (a) {\displaystyle f(a_{n})\rightarrow f(a)}. This concludes the proof.

In real analysis, for the more concrete case of real-valued functions defined on a subset E ⊂ R {\displaystyle E\subset \mathbb {R} }, that is, f: E → R {\displaystyle f:E\rightarrow \mathbb {R} }, a continuous function may also be defined as a function which is continuous at every point of its domain.