Week 6: Limits and Continuity. The derivatives

Let  be a function defined at all values in an open interval containing , with the possible exception of  itself, and let  be a real number. If all values of the function  approach the real number  as the values of  approach the number , then we say that the limit of  as  approaches  is . (More succinct, as  gets closer to  gets closer and stays close to .) Symbolically, we express this idea as

.

A function f(x)f(x) is said to be continuous at x=ax=a if

limx→af(x)=f(a)limx→a⁡f(x)=f(a)

A function is said to be continuous on the interval [a,b][a,b] if it is continuous at each point in the interval.

The difference between the continuous and differentiable function is that the continuous function is a function, in which the curve obtained is a single unbroken curve. It means that the curve is not discontinuous. Whereas, the function is said to be differentiable if the function has a derivative.