# Continuity leading to closure

jaci55555
The graph of a continuous funtions (R -> R) is the subset G:={(x, f(x) | x element of R} is a subset of R^2. Prove that if f is continuous, then G is closed in R^2 (with euclidean metric).

I know that continuity preserves limits, so xn -> x in X means f(xn-> y in Y.
and for all A element of R^2 - G there exists r > 0 st B(a, r) subset of R^2 - G.I know that if R^2 - G is open then G is closed.

What is the connection between limits and the open ball B? I think that might be the way...

## Answers and Replies

Staff Emeritus
Science Advisor
Homework Helper
the following characterization might be useful:

A subset F of a metric space X is closed iff for all sequences (x_n) in F which converges to a certain x in X, it holds that x is in F.