Definition of Continuity

Question

The definition of continuity I am using is the following:

Let $f$ be a real function, $a\in D(f)$. If for any sequence $\{x_n\}$ in $D(f)$ converging to $a$: $$\lim f(x_n)=f(a)$$ then $f$ is continuous at the point $a$.

Now there's one detail that bothers me. The text I am using (lecture notes) then briefly mentions:

Notice that we can only consider monotonic sequences

My question is simple: why only monotonic?

Thanks.

Answer 1

Assuming the domain of the function is a subset of $\mathbb R$ (otherwise it is not clear what monotonous should mean) I think the remark should be "Notice that it suffices to consider only monotone sequences". Clearly, if the condition holds for any sequence that it holds for a monotone one and so the the more general condition implies the one for monotone sequences. On the other hand, any sequence in $\mathbb R$ contains a monotone subsequence and you can use this to show that the more restrictive condition implies the more general one. Thus they are equivalent.

Answer 2

Because you can always take a monotone sequence from an arbitrary sequence. So if for some sequence ${x_n}$ in D(f) converging to a : $limf(x_n )$ not converges to f(a), then there is a subsequence $y_n$ in $x_n$, such that $|f(y_n)-f(a)|$>r>0, then take a monotone subsequence of ${y_n}$ you get that there is a monotone sequence such that the condition is not satisfied, this shows that you just need to consider the monotone sequence.