I've recently started learning about topology, and I was able to prove that any function which is undefined on a countable subset of $\mathbb R$ is also not continuous. However, I am struggling to prove the same claim for a function which is undefined on an uncountable subset of $\mathbb R$. Here's my proof for the countable case:
Denote the set of points upon which $f$ is undefined as $A$. Then, the preimage of the empty set is $f^{-1}(\emptyset) = A$. If $f$ is continuous, then $A$ should be open (as $\emptyset$ is open). However, the singleton set is not open with respect to the usual topology over $\mathbb R$, and since $A$ is a countable union of singleton sets it must also not be open. Hence, the preimage of an open set is not necessarily open, meaning $f$ is not continuous over $\mathbb R$.
This proof breaks down for me, because if $A$ is uncountable it can possibly be open. Taking the function defined by $f(x) = \exp\Big(-\frac{1}{\sqrt{x^2-1}}\Big)$ as an example, the corresponding set $A$ for this $f$ is $A = (-1, 1)$. In this case, $f^{-1}(\emptyset) = (-1, 1)$ is open, so that means $f$ does take open sets to open sets… and so $f$ is continuous.
Can anyone please explain the flaw in my logic?
Best Answer
It seems that your understanding of the concept of function is not the standard one. A function $f : X \to Y$ assigns to each input $x \in X$ a unique output $f(x) \in Y$.
Therefore the notation $f : \mathbb R \to \mathbb R$ implies that $f$ is defined on all of $\mathbb R$. There is nothing like a function $f : \mathbb R \to \mathbb R$ which is undefined on a nonempty subset of $\mathbb R$. What you have in mind is sometimes called a partial function. See here and here. But if you consider a partial function $f : X \to Y$ between topological spaces $X,Y$, you must be aware that it is actually a function $f : X ' \to Y$ defined on some $X' \subset X$. The continuity of $f$ has nothing to do with the points of $X \setminus X'$.
However, denoting certain partial functions as functions is well accepted in some contexts. In complex analysis (which does probably not belong to the field of your experience at the moment) one uses the concept of meromorphic functions on open subsets of the complex plane. Meromorphic functions are in fact partial functions in the strict sense.