Epsilon-Delta Continuity Visualizer

The rigorous definition of a limit, drawn as a box. Continuity at a point $x_0$ says: for every tolerance $\varepsilon$ you demand on the output, there is a tolerance $\delta$ on the input that delivers it, $|x - x_0| \lt \delta \Rightarrow |f(x) - f(x_0)| \lt \varepsilon$. Pick $\varepsilon$ and watch the widest $\delta$ that works appear: the curve over the $\delta$-interval stays trapped inside the $\varepsilon$-band, the graph leaving the box only through its sides, never its top or bottom. Shrink $\varepsilon$ and $\delta$ shrinks too, but it never has to reach zero, that is continuity. At a jump it does collapse: once $\varepsilon$ is smaller than the gap, no $\delta$ works at all. The diagnostic plots that response $\delta(\varepsilon)$.

Figure 1. Epsilon-delta continuity. Top: the graph, the ε-band around f(x₀), and the widest δ-interval whose image stays inside it, drawn as a box a moving test point never escapes. Bottom: the largest δ as a function of ε, which stays positive for a continuous point and collapses to zero at a jump. Method: bisection for the maximal δ.

x₀0.60

epsilon0.40

WHAT TO TRY

Shrink epsilon: the box narrows and the widest delta shrinks with it, but stays positive. The moving point never leaves the box.
Switch to the jump and set x₀ = 0: for small epsilon no delta works, the box cannot be drawn, and δ(ε) reads zero. That is a discontinuity.
On the jump, raise epsilon above the gap (about 1.2): suddenly a delta exists again, and the δ(ε) curve lifts off zero.
On the parabola at x₀ = 0 (a flat vertex) delta grows like √ε, slower than the straight-line case where it grows like ε.