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 function f : X→Y is continuous, then f⁻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.