Is the Function Defined by the Infimum of Distances Continuous?

[tylerc1991]

Homework Statement

Let (X,d) be a metric space and let A be a nonempty subset of X. Define a function f:X -> R^1 by f(x) = inf{d(x,a) : a is an element of A}. Prove that f is continuous.

Homework Equations

The Attempt at a Solution

Intuitively I can see that the function is continuous because it seems like for an arbitrary open interval in R^1 there is some pre-image of the function that is an open subset of this open interval, I just don't exactly know where to begin writing this. Can someone help me with the intuition behind this problem and let me know if I am on the right track? Thank you very much!

 

[ystael]

If you don't know how to attack a continuity problem using the abstract (preimage of an open set is open) definition, try instead to prove that the function is continuous at any arbitrarily chosen point using the good old epsilon-delta definition.

 

[losiu99]

In both cases, you may want to prove that
$$|f(x)-f(y)|\leq d(x, y)$$