I wanted to discuss my intuition of why we formulate the concept of continuity in terms of pre-image of open set is open instead of images for example if we consider $f(x) = c$ where $c$ is some constant, then that should be continuous, but if we formulated continuity in terms of images it won't be continuous. Is the reason we do it that way, because pre-images will always guarantee that function in the domain can only have $1$ value?
Topology – Why Formulate Continuity in Terms of Pre-images Instead of Image?
continuitygeneral-topology
Best Answer
It's not nice to formulate it in terms of images. The behavior of images varies wildly between various continuous functions. Take the interval $(-10,10)$ and the functions $f_1(x) = x$ and $f_2(x) = \sin x$. Then the image of $(-10,10)$ under $f_1$ is of course $(-10,10)$, but the image of $(-10,10)$ under $f_2$ is $[-1,1]$. With one function you get an open set, with another you get a closed set. You can of course explore the other cases of half-open/half-closed or other phenomena. You can't make any meaningful statement about images of open sets. You can't even say anything for certain about local behavior (the images of "small" open sets) because there are plentiful locally constant functions.
You might want to flip this on its head then since images of open sets doesn't work: what about images of closed sets? This also doesn't work. Consider $\mathbb{R}$ and the function $f_3(x) = \arctan x$, then the image of $\mathbb{R}$ under $f_3$ is $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ which is an open set.
The beauty of pre-images is that the functions we intuitively look at as being continuous have open sets as pre-images of open sets. Perhaps the closest you can get to a statement about images is that continuous functions map compact sets to compact sets, however there are discontinuous functions which also do this: consider $[-1,1]$ and $f_4(x) = 1$ if $x>0$, $f_4(0) = 0$ and $f_4(x) = -1$ if $x<0$. Then $f_4$ maps a compact set to a compact set but it is definitely not continuous in the usual topology.