Continuous function

In mathematics a function is said to be continuous if a small change in the inputs only cause a small change in the output. If this is not the case, the function is said to be discontinuous. Functions defined on the real numbers, with one input and one output variable will show as an uninterrupted line. They can be drawn without lifting the pen. The definition given above was made by Augustin-Louis Cauchy.

Karl Weierstrass gave another definition of continuity: Suppose again that there is a function, defined on the real numbers. At the point $x_{0}$ the function will have the value $f(x_{0})$. If there is a value $\varepsilon >0$ and a value $\delta >0$ so that $|x-x_{0}|<\delta$ and $|f(x)-f(x_{0})|<\varepsilon$ for all $x$, then the function is continuous in $x_{0}$. 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 $x_{0}$