Continuous function

A mathematical function is called continuous if, roughly said, a small change in the input only causes a small change in the output. If this is not the case, the function is discontinuous. Functions defined on the real numbers, with one input and one output variable, will show as an uninterrupted line (or curve). They can be drawn without lifting the pen off of the page. The definition given above was made by Augustin-Louis Cauchy.^[1]

Karl Weierstraß gave another definition of continuity: Imagine a function f, defined on the real numbers. At the point ${\displaystyle x_{0}}$ the function will have the value ${\displaystyle f(x_{0})}$. If the function is continuous at ${\displaystyle x_{0}}$, then for every value of ${\displaystyle \varepsilon >0}$ no matter how small it is, there is a value of ${\displaystyle \delta >0}$, so that ${\displaystyle |x-x_{0}|<\delta }$, means that ${\displaystyle |f(x)-f(x_{0})|<\varepsilon }$. We can put this another way, given a point close to ${\displaystyle x_{0}}$ (called x), the absolute value of the difference between the two values of the function can be made increasingly small, if the point x is close enough to ${\displaystyle x_{0}}$.

There are also special forms of continuous, such as Lipschitz-continuous. A function is Lipschitz-continuous if there is a ${\displaystyle L}$ with ${\displaystyle |f(x)-f(y)|\leq L|x-y|}$ for all x,y ∈ (a,b).

One kind of discontinuous function

A basic way to know if a function is continuous is to use a pencil or your finger. Then, start at the left of the function. Then, move your finger along the path of the function. If you ever need to lift your finger or pencil to keep following the function, then you know it is not continuous. This is because, by lifting your finger, you have "jumped" from one section of the function to another. That means you made a very small movement but the function changed very much. This is what the first sentence of this article is talking about.

Related pages

• Calculus
• Chaos theory

References

1. Fischer, Helmut; Helmut Kaul (2007). Mathematik für Physiker Band 1: Grundkurs. Teubner Studienbücher Mathematik. Teubner. p. 165 ff. ISBN 978-3-8351-0165-4.