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How can a graph show that a function is continuous at a point?

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transformation of a circular region into an approximately rectangular region — analysis \
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr². As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr²—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

In mathematics, the intuitive concept of a function being continuous at a point is often described as a graph that can be drawn without any abrupt breaks, jumps, or gaps.12

Visually, a function is continuous at a point $x_{0}$ if the graph passes through that point smoothly. If you were to trace the curve with a pencil, you would not need to lift it from the paper to continue the line at that location. Conversely, if a graph exhibits a sudden jump—such as a function that equals 0 for $x \leq 1$ and 2 for $x > 1$ —it is not continuous at $x = 1$ because the graph is broken at that point.1

Formally, in analysis, this visual intuition is defined by limits: a function $f$ is continuous at a point $p$ if the limit of $f (t)$ as $t$ approaches $p$ is equal to the actual value of the function at that point, $f (p)$ .2