Distance to Set is Continuous

An ε-δ argument for uniform continuity

Hi Scholars!

This week we revisit a classic example about the continuity of distance functions. The key intuition originates from drawing 3 points and an example “blob” of a set. The points x and y outside the set and a point p in the set. Our goal is to relate |f(x)-f(y)| to the distance d(x,y). Since our function is defined as an infimum, we know f(x) ≤ d(x,p) and f(y) ≤ d(y,p). And, by the triangle inequality, we know d(x,p) ≤ d(x,y) + d(y,p). Rearranging these and using the fact an infimum is the greatest lower bound, we find |f(x) - f(y)| ≤ d(x,y). Then one can pick δ=ε to complete the proof.

Stay Awesome.

Howard

p.s. To see a complete proof, check out the slides below.