Week 6: Limits and Continuity. The derivatives

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

\underset{x\to a}{\lim}f(x)=L.

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


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.