Continuity

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Intuitively, a continuous function is one whose graph has no sudden jumps in it; the graph is all a single connected piece. Such a function can be drawn without picking the pen up off of the paper. Formally, a function  is defined to be continuous if for any real and any infinitesimal , is infinitesimal.

Example

Let the function  be defined by for , and for . Then f (x ) is discontinuous, since for dx > 0, f (0 + dx ) − f (0) = 1, which isn’t infinitesimal.
 

Figure 3.1 Caption

The black dot indicates that the endpoint of the lower ray is part of the ray, while the white one shows the contrary for the ray on the top

 

If a function is discontinuous at a given point, then it is not differentiable at that point. On the other hand, the example shows that a function can be continuous without being differentiable.

In most cases, there is no need to invoke the definition explicitly in order to check whether a function is continuous. Most of the functions we work with are de- fined by putting together simpler functions as building blocks. For example, let’s say we’re already convinced that the functions defined by and are both continuous. Then if we encounter the function , we can tell that it’s continuous because its definition corresponds to. The functions and have been set up like a bucket brigade, so that takes the input, calculates the output, and then hands it off to for the final step of the calculation. This method of combining functions is called composition. The composition of two continuous functions is also continuous. Just watch out for division. The function is continuous everywhere except at , so for example is continuous everywhere except at multiples of , where the sine has zeroes.