# Continuous Function

Continuous means without any break or sudden jump or holes at any point. In Mathematics, continuous function also has similar kind of definition. If a small change in the input of a function brings a small change in the output of that function, then we say that the function is continuous. For the function to be continuous at any point x=a, the function must be defined at that point and limiting values of f(x) when x approaches a, is equal to f(a). In other words we say, If f(x) is a function such that $\lim_{x \rightarrow a}$ f(x) = f(a), then f(x) is continuous at “a”. **Weierstrass definition (epsilon-delta definition) of continuous functions:** A function ƒ is said to be continuous at the point c, if for any number $\epsilon$ > 0, however small, there exists some number $\delta$ > 0 such that for all x in the domain of ƒ with |x-c| < $\delta$, the value of ƒ(x) satisfies |f(x) –f(c)| < $\epsilon$.Some real life examples for continuous functions are: Temperature at various times of the day, height growth as a function of time, cost of the cab ride as a function of distance travelled, population growth in a city as a function of time, etc.

If f(x) is not continuous at x = a, we say that f(x) is discontinuous at x = a.