Question: Suppose $f(x)$ is periodic with period p and $g(x)$ is periodic with period q. Let $r$ be the L.C.M. of p and q, if it exists. Then show that:

1. If $f(x)$ and $g(x)$ cannot be interchanged by adding a least positive number $k<r$, to $x$, then $r$ is the period of $f(x)+g(x)$.
2. If $f(x)$ and $g(x)$ can be interchanged by adding a least positive number $k<r$, to $x$, then $k$ is the period of $f(x)+g(x)$.

I know I should give my approach towards the problem but I can't seem to be able to make the head or tail of this problem. I don't think I even completely understand what's being said.

Answer: 
A topological property is any property that is preserved by all homeomorphisms. A homeomorphism from one metric space $X$ to another, $Y$, is a function $f:X\to Y$ such that

* $f$ is continuous;
* $f$ is one-to-one;
* $f^{-1}$ is continuous;
* $f$ is onto, i.e. every point in $Y$ is the image of some point in $X$.

Suppose one defines a "purple" metric space. To say "purpleness" is a "topological property" means that for every metric space $Y$ for which a homeomorphism from $X$ to $Y$ exists is also purple.

So your task would be: **either**

* Show that whenever a function $f$ exists with the properties above and $X$ is "purple", then the space $Y$ must also be "purple", then conclude that "purpleness" is a topological property of metric spaces; **or**
* Show that some metric space $Y$ and some function $f$ with the properties above exists and that $X$ is "purple" and $Y$ is **not** "purple", then conclude that "purpleness" is **not** a topological property.

So let's consider your first question:

* If every continuous function $g:X\to\mathbb R$ is bounded and $f:X\to Y$ is a homeomorphism from the metric space $X$ to some other metric space $Y$, is that enough to prove that every continuous function $h:Y\to\mathbb R$ is bounded? If so, then the property involved is topological.
* But if there is some metric space $Y$ and some function $f$ with the propertires above, and some function $h:Y\to\mathbb R$ that is *not* bounded, then the property involved is *not* topological.
In short, a property of a metric space is topological if it does not (really) depend on the metric. That is, if you have a different metric that produces the same notions of convergence, continuous functions, etc. (or even if you *only* have the notions of convergence, contnuou functions, etc.) then the property does not change.

Note that $b$ and $c$ can be greatly simplified to a truth value that holds completely independently of the metric $d$; which makes at least these properies topological.