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How does continuity describe a function without abrupt jumps?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

In mathematics, continuity provides a rigorous way to describe a function that varies without abrupt breaks or jumps. Intuitively, a function is continuous if small changes in the input ( $x$ ) result in only small changes in the output ( $y$ ).12

To define this precisely, a function $f (x)$ is continuous at a point $x_{0}$ if, for any desired degree of closeness ( $ϵ$ ) for the output values, one can find a corresponding distance ( $δ$ ) for the input values such that all $x$ within that distance from $x_{0}$ produce $y$ -values within the desired closeness to $f (x_{0})$ .1

If a function lacks this property, it is considered discontinuous. For example, a function that jumps from 0 to 2 at a specific point is discontinuous because the difference between the function's value at that point and values immediately adjacent to it remains significant, failing the requirement that the output difference becomes arbitrarily small as the input difference approaches zero.12