continuity functions

A function is said to be continuous if and only if it is continuous at every point of its domain. A function is said to be continuous on an interval, or subset of its domain, if and only if it is continuous at  each point of the interval. The sum, difference, and product of continuous function with the same  domain  are additionally continuous, as is the quotient, except at points at which the denominator is zero. Continuity can  be defined in terms of limits by saying that f(x) is continuous at x0 of its domain if and only if, for  values of x in its domain

Consider the graph of f(x) = x3 − 6x2 − x + 30:

x3 

We can see that there are no "gaps" in the curve. Any value of x will give us corresponding  value of y. We can continue  the graph in the negative and positive directions.Such functions are known as continuous  functions.

A function ƒ : X → Y is a uniform continuity if  for every real number ε > 0 there exists δ > 0 such that d₂(ƒ(x), ƒ(y)) < ε for every x, y ∈ X with d₁(x, y) < δ.