The thermostat's promise
A good thermostat promises: name any comfort window you want (epsilon), and it can set a dial precision (delta) that keeps the room inside it. Epsilon-delta continuity is exactly that promise — small enough input wiggles keep the output as close as you demand.
Small input, small output
f is continuous at a if for every ε > 0 there exists δ > 0 such that |x − a| < δ implies |f(x) − f(a)| < ε. In words: you can force the output as close to f(a) as you like by keeping the input close enough to a.
A jump breaks it
At a jump discontinuity, outputs leap by a fixed gap. For ε smaller than that gap, no δ works — the function is discontinuous there.
Lab · Hunt for delta
- Pick f(x) = 2x and the point a = 3, so f(a) = 6.
- Choose ε = 0.1. You need |2x − 6| < 0.1.
- Solve for how close x must be to 3.
What you should see: |2x − 6| = 2|x − 3| < 0.1 means |x − 3| < 0.05, so δ = 0.05 works. A δ always exists, so f is continuous.