Example 6.2.4(c):
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 D, but the sequence 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 ?