Continuity at a point means a function behaves predictably as you approach a specific value.
Mechanistically, a function f is continuous at x = c if three conditions are met: f(c) exists, the limit of f(x) as x approaches c exists, and the limit equals f(c).
This definition is not just a formality; it's the backbone of calculus.
Without continuity, derivatives and integrals lose their meaning.
The mechanism behind this is simple: if you can draw the function without lifting your pen at x = c, it's continuous there.
The challenge is in the detail.
You might think a function is continuous just because it looks smooth, but a hidden hole or jump can trip you up.
Misinterpreting the conditions — especially the limit condition — leads to mistakes.
You might assume continuity based on appearance rather than checking if the mathematical criteria are satisfied.
That error costs you when justifying conclusions about a function's behavior.