Discontinuity
A discontinuity is a point where a function breaks, jumps, or has a hole, making it impossible to draw without lifting your pencil.
Definition
A discontinuity is a place on a graph where there is a hole, a jump, or a vertical gap, a spot where the function breaks down or behaves unexpectedly. There are three main types: removable (a hole where the limit exists but $f(a)$ is missing or wrong), jump (left and right limits exist but are unequal), and essential or infinite (at least one one-sided limit fails to exist finitely, often because the function grows without bound or oscillates near the point).
Example
If a taxi meter reads $\$5$ up to $1$ mile, then jumps instantly to $\$8$ for $1.01$ miles, there is a jump discontinuity at $1$ mile. Algebraically: $f(x) = (x^2-1)/(x-1)$ at $x=1$ is removable, $\lfloor x \rfloor$ at every integer is a jump, and $f(x) = 1/x$ at $x=0$ is infinite. A more extreme case is $f(x) = \sin(1/x)$ near $x = 0$: it oscillates infinitely often, so neither one-sided limit exists, making it an essential discontinuity.
Key Insight
Discontinuities mark the places where a mathematical model might fail to reflect smooth, real-world change, and identifying the type tells you whether the break can be "fixed" (removable) or is fundamental to the function's behavior. This classification underpins the Lebesgue characterization of Riemann-integrable functions: $f$ is Riemann integrable on $[a,b]$ if and only if its set of discontinuities has measure zero.