## 6.2. Continuous Functions

### Example 6.2.4(c):

*f(x)*is continuous in a domain

*, and*

**D***{x*is a Cauchy sequence in

_{n}}*, is the sequence*

**D***{f(x*also Cauchy ?

_{n})}*f*is continuous at

*c*, and

*{x*is a sequence converging to

_{n}}*c*, then

*f(x*must converge to

_{n})*f(c)*. And since convergent sequences are Cauchy one would assume that the statement should be true.

But of course life is not so simple. Consider the function
*f(x) = 1/x* and the sequence *{1/n}*. Then the sequence
is convergent to zero, and thus is Cauchy. But *f(1/n) = n*, which is not
Cauchy. This function is continuous in the domain *D = (0, 2)*,
say, the sequence

*{1/n}*is Cauchy in

*, but the sequence*

**D***f(1/n)*fails to be Cauchy.

The point here is that the function does not need to be continuous
at the limit point of a sequence, and hence the above statement
is false. While the sequence *{1/n}* converges to zero, the function
*f(x)* is not continuous at zero. Yet the sequence *{1/n}* is Cauchy
in *(0, 2)*.

Is the statement true if the domain of the function is all of
* R*? Can you find other formulations for which the statement
would become true ?