This is an article I wanted to write for a long time.

What exactly is topological continuity? A mapping $f:X\to Y$ between two topological spaces is said to be continuous if for any open set $U\subset Y$, $f^{-1}(U)$ is open in $X$.

After reading this definition, very few students understand the motivation behind this definition. By bringing these concepts to Euclidean space, a  picture slowly begins to form. This definition is valid in $\Bbb{R^n}$. However, what was the need to unnecessarily generalize continuity to non-Euclidean spaces? What does continuity really mean in a general setting?

Given below is my impression of continuity, and has not been influenced by any mathematical work that I have come across.

If there is a continuous mapping from $X$ to $Y$, this means that $Y$ is at least as “solid and smooth”, or more “solid and smooth” than $X$. I have no intention of making the phrase “solid and smooth” precise, and it is an arbitrary phrase concocted by me to give readers a “feel” for the nature of continuity. As an intuitive explanation, a circle is more “solid and smooth” then a line, which has two sharp points. The graph of $y=x$ is more “solid and smooth” than $y=\lfloor x\rfloor$. Developing more examples of this kind would be a useful exercise. The less the number of “sharp points”, the more an object is “solid and smooth”.

Now we swoop back to my original claim: that if $f:X\to Y$ is continuous, then $Y$ is at least as or more “solid and smooth” than $X$. As an example, consider the continuous mapping of $[0,1]$ to $\Bbb{S}^1$. The continuous map for this would be $t\to (\cos 2\pi t,\sin 2\pi t)$. We know that $[0,1]$ has two “sharp points” (at 0 and 1), while $\Bbb{S^1}$ has none. Hence, $\Bbb{S}^1$ is at least as or more solid and smooth than $[0,1]$. However, there is no continuous mapping from $\Bbb{S}^1$ to $[0,1]$, as the latter is less solid and smooth than the former.

It is a useful exercise to use this concept in more complicated topological cases.

Now we come to path homotopy. When we say there is a continuous mapping from $[0,1]\times [0,1]$ to a topological space such that $f(I\times \{0\})=C_1$ and $f(I\times \{1\})=C_2$, then we are in essence saying that the space encompassed and bordered by $C_1$ and $C_2$ is as or more solid and smooth then $I_0^2$. Hence, there are no breaks in between. It can be pictured as a solid block of ink. No spaces.

This is the reason why $[0,1]$ is generally used as the domain for a continuous map. We want the image to be solid. No gaps.