Examples of continuous functions that are not homotopic to each other. Let $f,g:X \to Y$ be continuous functions. Then, consider $F:X \times I \to Y$ defined as follows:$$F(x,t) = f(1-t)+gt$$
How does one check whether $F$ is continuous(w.r.t the product topology)?
Is there any other way to define a homotopy between two functions?
As an example, take $f,g :  \mathbb{R} \to \mathbb{R}$ to be $f(x) = x, g(x) = sin(x)$. Then how does one PROVE that the homotopy map as defined above is continuous?
Moreover, what are some examples of maps that are not homotopic?

The identity map of the circle $\rm{id}:S^1\to S^1$ and a constant map on $S^1$ are continuous but not homotopic.  This is due to a result by Heinz Hopf, from one point of view.
Intuitively the unit circle is not contractible.
By the alluded to result, we can wrap the circle around itself as many times as we like, continuously, and get non-homotopic maps.

The homotopy you refer to is the straight line homotopy.  There are plenty of others.
