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**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→af(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.