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Roughly speaking: a function is continuous if we can draw its graph without lifting our pencil from the paper.  This "definition" makes little sense from a mathematical point of view, so we investigate it with greater detail.

First of all the formal definition:

Given a function img, and a point c in the domain D, the function f is said to be continuous at c if either c is not an accumulation point for D, or, when c is an accumulation point, img.

Important remarks

The point c must be in the domain of the function.
The value l that we used in the definition of limit is now replaced by f(c).

Unlike the case of limits, here we do not want to investigate the behaviour of a function as x gets closer and closer to a given point c; on the contrary we want to check if the function approaches exactly the value f(c) as x approaches c.

Observe that, if c is a left (right) end point in the domain, we need to check only img (img).

It's easy enough to prove that all elementary functions are continuous whatever point you take in the domain. So in order to find interesting examples we'll often use other kinds of functions, mainly functions defined in pieces.

graph of xsin(1/x)

Facts about continuous functions

As continuity is defined in terms of limits, it's obvious that sums, products, quotients and compositions of continuous functions are continuous. 

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first published on march 26 2002 - last updated on september 01 2003