Continuity

$f$ is continuous at $a$ if $\lim_{x\to a} f(x) = f(a)$: small changes in the input produce small changes in the output. Continuous on an interval means you can draw the graph without lifting the pen.

Measurements are never exact. If a model is continuous, a slightly wrong input gives a slightly wrong output, and computing with approximations makes sense. Continuity is also the hypothesis that unlocks the existence theorems (Bolzano, Weierstrass) on which root-finding and optimization rely.

Three things must hold at $a$: $f(a)$ is defined, the limit exists, and they agree. Each failure is a kind of discontinuity: a hole, a jump, or a wild oscillation. Geometrically, zooming in on a continuous graph never reveals a gap.

Sums, products, quotients (where the denominator is non-zero) and compositions of continuous functions are continuous; so are polynomials, $e^x$, $\sin$, $\cos$, and $\ln$ on $(0, \infty)$.

The step function $H(x) = 1$ for $x \ge 0$ and $0$ otherwise is discontinuous at 0. The perceptron used it as its activation, and that is precisely why gradient methods could not train multilayer perceptrons until the step was replaced by the continuous (and differentiable) sigmoid.

Continuity is the minimal promise that a numerical method can work: rounding the input should not change the answer much. Where models are discontinuous (decision boundaries, collisions in a physics engine, edges in an image) software needs special handling.