# Relating ε-δ Continuity to Open Inverses

### A proof of one direction of the equivalence

Hi Scholars!

This week during office hours we talked about the following lemma, which is one half of a theorem showing the equivalence between the ε-δ definition of continuity and a definition with inverses of a function.

Lemma: If a functionf: X→Y is continuous, thenf⁻¹(V) is open for each open set V⊆Y.

The perhaps tricky part of proving this result is jumping back and forth between *f⁻¹*(V) and V when applying the hypotheses. The graphic above steps through my thought process when thinking about this problem. Here is a rough sketch.

Start with what must be shown; in this case, we must verify a given point

*p*is an interior point of*f⁻¹*(V).Apply our knowledge that V is open to jump to V and note

*f*(*p*) is an interior point of V. This means there is an ε-ball about*f*(*p*) in V.Jump back to

*f⁻¹*(V) and apply our other hypothesis, continuity of*f*, to get a “special” δ-ball about*p*for which applying*f*gives something inside the ε-ball about*f*(*p*), which is in V (by Step 2).This implies the δ-ball about

*p*is inside*f⁻¹*(V), completing the proof.

That’s all for this week. As always, let me know when you come across any concepts best clarified via a nice diagram.

Stay Awesome.

Howard

p.s. If you prefer a static image, below is a copy of that version.