## definition of continuity

### In mathematics, continuity is a fundamental concept that describes the behavior of a function

In mathematics, continuity is a fundamental concept that describes the behavior of a function. A function is said to be continuous if, intuitively, its graph has no “jumps,” “holes,” or “breaks.”

More formally, let’s consider a function f(x) defined on a subset of the real numbers. We say that f(x) is continuous at a point c if three conditions are met:

1. Existence of the function at c: The function f(x) is defined at c, meaning it is defined and has a finite value at that point.

2. Limit existence at c: The limit of f(x) as x approaches c exists and is equal to the value of f(c).

3. Limit continuity at c: The limit of f(x) as x approaches c is equal to the value of f(c).

If a function satisfies these conditions for every point in its domain, then we say that the function is continuous on its entire domain.

There are different types of continuity that we often encounter:

1. Pointwise continuity: A function f(x) is said to be pointwise continuous at a point c if it satisfies the three conditions mentioned above.

2. Uniform continuity: A function f(x) is said to be uniformly continuous on an interval if, for any two points within that interval, the difference in function values can be made arbitrarily small by choosing the corresponding differences in the input values to be small enough.

3. Continuity on a closed interval: A function f(x) is said to be continuous on a closed interval [a, b] if it is continuous at every point within that interval, including the endpoints a and b.

Continuity is a key concept in calculus and analysis, as it allows us to define and study important properties of functions, such as derivatives and integrals. It plays a crucial role in understanding the behavior of mathematical models and real-world phenomena.

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