# Continuous function

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).
1. Fischer, Helmut (2007). Mathematik für Physiker Band 1: Grundkurs. Teubner Studienbücher Mathematik. Teubner. p. 165 ff. ISBN 978-3-8351-0165-4. Unknown parameter |coauthors= ignored (|author= suggested) (help)