Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. Define the function $F:\mathbb{R}\rightarrow \mathbb{R}$ as $$F(x)=f(x)\prod_{n=1}^\infty\frac{x-\frac{1}{n}}{x-\frac{1}{n}}$$
I have several questions regarding the above, and similarly constructed functions. Intuitively, the function $F(x)$ is almost the same function as $f(x)$. But, there are countably many singularities which have an accumulation point at $f(0)$. The function $F$ also retains the limit structure of $f$ (i.e. $\lim\limits_{x\rightarrow y}F(x)=\lim\limits_{x\rightarrow y}f(x) = f(y)$).
I have not had an in depth study of integration. Although, since the set of $x$ such that $F(x)$ is not defined has measure $0$, I would expect some form of integration to be consistent for all intervals of the real line.
So the singularities of $F$ seem trivial. However, I would argue the function $F$ is strictly different than the function $f$. Since the construction of $F$ required the knowledge of the continuous function $f$, it seems clear that we can, in some manner, "patch" the holes, and treat this function as $f$.
Question 1: Are there certain properties of $F$ which would intuitively be the same as $f$, but are not the same under comparison? (I understand that, looking at $F$ and $f$ as a 1-manifold, there are certainly many topological properties which are not consistent, but this is attributed to $F$ being discontinuous. I'm looking more for an analysts perspective here).
Question(s) 2: If we looked instead at functions of a complex variable, does the $F$ lose any of the information contained in $f$? I mean to say, if $f$ is a continuous and analytic function, then would integrating around the closed contour $|z|=2$ be the same as for $F$? What must be true to about the singularities for this to change?
I apologize if I am vague. I'm not looking for answers to each question. I've just been thinking about this and wondering if there is any information regarding functions like the one above. If I am misunderstanding anything, I would appreciate being straightened out and have see my ideas written formally. To the same extent, don't go easy on me if you can explain the topic!