*Kiddle Encyclopedia.*

# Continuous function facts for kids

In mathematics a function is said to be **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 said to be **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*. The definition given above was made by Augustin-Louis Cauchy.

Karl Weierstraß gave another definition of continuity: Suppose that there is a function *f*, defined on the real numbers. At the point the function will have the value . If the function is continuous at , then for every value of no matter how small it is, there is a value of , so that whenever , that makes . Intuitively: Given a point close to (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 .

There are also special forms of continuous, such as *Lipschitz-continuous*. A function is Lipschitz-continuous if there is a with for all x,y ∈ (a,b).