continuity = topology. study the first topological result in calculus, the intermediate value theorem, and think about its proof.
there are some helpful comments in this wiki article, including links between IVT and topological "connectedness".
https://en.wikipedia.org/wiki/Intermediate_value_theorem
one usefulness of topology is to make arguments easier for existence of solutions to problems when explicit solutions are hard to find. e.g. if f is a function and we want to find x such that f(x)=0, it may be very hard to find such an x. But by topology, if we can find an a with f(a)<0 and b with f(b)>0, AND if we can show f is defined and continuous on the interval [a,b], then an x with f(x) = 0 is guaranteed to exist somewhere in the interval (a,b) even if we cannot find it.
a generalization of this principle to higher dimensions can be used to show that if f is a continuous map from the unit disc to the plane, which maps each point of the boundary circle to itself, then again there is some point x in the open disc with f(x)=(0,0).
Similar generalizations exist for maps of n dimensional space, and no doubt also to infinite dimensional complete normed spaces, but I am not an expert on infinite dimensional analysis.
There is a topological argument that, in integral calculus, one cannot "rationalize" the integral of dx/sqrt(1-x^4). The analogous integrand dx/sqrt(1-x^2) can be rationalized by the substitution x = 2t/(1+t^2), but it can be shown that any such change of variables rationalizing the first integrand above would yield an orientation preserving continuous branched covering of the torus by a sphere, which is impossible by computing topological Euler characteristics. I.e. for such maps, Euler characteristics cannot go down, but the euler characteristic of the sphere is 2 and that of the torus is zero.
for an elementary example linking analytical and topological concepts, you might try proving that is f is a real valued function defined on the real line, with f(0) = 0, then:
1) the inverse image of every open interval containing 0, contains an open interval containing 0,
if and only if
2) every sequence converging to 0, maps under f to a sequence converging to 0.
I.e. show the calculus definition of continuity is equivalent to the topological space definition. This sort of thing is presumably in Simmons.
I think I got my own first sense of clarity from Simmons in reading a proof that every bounded increasing sequence of reals must have a limit, another abstract "existence proof", where we prove something of interest exists without actually finding it.
Another beautiful example is the contraction lemma or Banach fixed point theorem in metric spaces, that implies the existence of solutions of first order ordinary differential equations under certain hypotheses. The following link discusses this and other natural results in analysis such as the inverse function theorems in calculus that follow:
https://en.wikipedia.org/wiki/Banach_fixed-point_theorem
after connectedness, the other most important topological concept for analysis is "compactness". a subset of the real line is compact iff it is closed and bounded, e.g. a closed finite interval [a,b] is compact. then every real valued function defined and continuous on a compact domain has a maximum (and minimum) value there. this is another existence result. then you use derivatives to try to actually find that maximum or minimum.
