Continuity

Continuity is a fundamental concept in mathematics and, more specifically, in topology. It describes the behavior of functions in a rigorous way and has applications in numerous areas of mathematics and beyond, including physics and engineering.

Definition

In the context of real numbers, a function f is said to be continuous at a point a if, roughly speaking, small changes in the input around a result in small changes in the output. More formally, f is continuous at a if for every positive number ε, there exists a positive number δ such that for all x with |x - a| < δ, we have |f(x) - f(a)| < ε.

Properties

Continuity has several important properties. For example, the sum, product, and quotient (provided the denominator is not zero) of continuous functions are continuous. The composition of continuous functions is also continuous.

Continuity in topology

In topology, the concept of continuity is generalized to functions between arbitrary topological spaces. A function f : X → Y between topological spaces is continuous if the preimage of every open set in Y is open in X.

Applications

Continuity is a key concept in many areas of mathematics and its applications. For example, in calculus, the intermediate value theorem, which states that a continuous function on a closed interval takes on every value between its minimum and maximum, relies on continuity. In differential equations, the existence and uniqueness of solutions to certain types of equations can be proven using continuity.

See also

• Differentiability
• Limit (mathematics)
• Uniform continuity

References

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