Convergence of a function at a point

Definition 3.4.1

Let (X, d_X) and (Y, d_Y) be metric spaces, f : E → Y, E ⊂ X, x₀ an accumulation point of E, and y₀ ∈ Y.

We say f converges to y₀ at x₀ if:

∀ε > 0, ∃δ > 0 such that ∀x ∈ E with 0 < d_X(x, x₀) < δ, we have d_Y(f(x), y₀) < ε

In plain English: No matter how small an ε-window you draw around y₀ in the output, I can always find a small enough δ-window around x₀ in the input so that every point of E inside the δ-window (except x₀ itself) gets mapped inside the ε-window. Try dragging the sliders below!

A simple parabola converging to 1 at x₀ = 1.

ε (epsilon)0.50

Output tolerance around y₀

δ (delta)0.40

Input window around x₀

Show ε/δ bandsShow sample points

How to use this visualization:
1. Pick an ε — this is the "challenge" (how close to y₀ must f(x) be?).
2. Your job is to find a δ so that all sample dots are green (meaning they land inside the ε-band).
3. If you can always do this — no matter how small ε gets — then f converges to y₀ at x₀.
Red dots = points that escape the ε-band. Shrink δ until they all turn green!