Discontinuity
Calculus & Advanced MathA 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. It is a spot where the function breaks down or behaves unexpectedly.
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.
Key Insight
Discontinuities mark the places where a mathematical model might fail to reflect smooth, real-world change.
Definition
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 infinite (the function grows without bound near a point, creating a vertical asymptote).
Example
Removable: $f(x) = (x^2-1)/(x-1)$ at $x=1$. Jump: $\lfloor x \rfloor$ at every integer. Infinite: $f(x) = 1/x$ at $x=0$.
Key Insight
Identifying discontinuity type tells you whether it can be "fixed" (removable) or whether the break is fundamental to the function's behavior.
Definition
A function has a removable discontinuity at $a$ if $\lim_{x \to a} f(x)$ exists but does not equal $f(a)$. It has a jump discontinuity if both one-sided limits exist but differ. An essential discontinuity (oscillatory or infinite) occurs when at least one one-sided limit fails to exist finitely.
Example
$f(x) = \sin(1/x)$ near $x = 0$ is an essential discontinuity: $f$ oscillates infinitely often, so neither one-sided limit exists.
Key Insight
The classification of discontinuities 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.