# Continuous function

Karl Weierstrass gave another definition of continuity: Suppose that there is a function f, defined on the real numbers. At the point $x_0$ the function will have the value $f(x_0)$. If the function is continuous at $x_0$, then for every value of $\varepsilon>0$ no matter how small it is, there is a value of $\delta > 0$, so that whenever $|x - x_0| < \delta$, that makes $|f(x) - f(x_0)| < \varepsilon$. Intuitively: Given a point close to $x_0$ (called x), the absolute value of the difference between the two values of the function can be made arbitrarily small, if the point x is close enough to $x_0$.
There are also special forms of continuous, such as Lipschitz-continuous. A function is Lipschitz-continuous if there is a $L$ with $|f(x) - f(y)| \le L|x - y|$ for all x,y ∈ (a,b).