# Thread: Continuous Functions on Metric Spaces

1. ## Continuous Functions on Metric Spaces

What is the definition of a continuous function on a metric space?

And what are the basic properties?

Thank you!

2. Originally Posted by lindsays
What is the definition of a continuous function on a metric space?

And what are the basic properties?

Thank you!

Definition 1: a function f :X --> Y between metric spaces X,Y with metrix d_X , d_Y is continuous at x in X if for any e > 0 there exists h > 0 s.t. that d_X(x,w) < h ==> d_Y(f(x), f(w)) < e.

Def. 2: a function f: X --> Y between metric spaces X,Y is continuous (in general) if the inverse image of any open ball in X is open in Y

Def. 3: a function F: X --> Y betwen metrix spaces...and etc. is continuous at x in X if for any open ball B in Y containing f(x) there exists an open ball A in X containing x s.t. f(A) is contained in B.

Choose the one you love the most or suits you better.

Tonio

3. Originally Posted by tonio
Definition 1: a function f :X --> Y between metric spaces X,Y with metrix d_X , d_Y is continuous at x in X if for any e > 0 there exists h > 0 s.t. that d_X(x,w) < h ==> d_Y(f(x), f(w)) < e.

Def. 2: a function f: X --> Y between metric spaces X,Y is continuous (in general) if the inverse image of any open ball in X is open in Y

Def. 3: a function F: X --> Y betwen metrix spaces...and etc. is continuous at x in X if for any open ball B in Y containing f(x) there exists an open ball A in X containing x s.t. f(A) is contained in B.
To be clear: def 1 above is the one found in most textbooks introducing metric spaces.
Def 2&3 are then proven a theorems.