the Casorati-Weierstrass theorem
/ kah-zoh-RAH-tee VY-er-shtrahss /
Near a pole a function does one tidy thing: it runs off to infinity. Near an essential singularity it does the opposite of tidy — it goes everywhere. The Casorati-Weierstrass theorem makes this vivid: no matter how small a neighbourhood of an essential singularity you zoom into, the function's values fill the complex plane so thickly that they come arbitrarily close to every single complex number.
Precisely: if z_0 is an essential singularity of f, then for any punctured neighbourhood 0 < |z - z_0| < delta, however tiny, the image f(of that neighbourhood) is DENSE in the complex plane. 'Dense' means: pick any target complex number w and any tolerance epsilon, and you can find a point z in that neighbourhood with |f(z) - w| < epsilon. So f does not avoid any region; it sprays its values everywhere, right up against every value. The proof is a clean argument by contradiction: if f stayed at distance at least epsilon from some w on the neighbourhood, then 1/(f(z) - w) would be bounded, hence have a removable singularity by Riemann, which would force f to have at worst a pole at z_0 — contradicting that z_0 is essential.
This theorem is the sharp dividing line between poles and essential singularities. A pole AVOIDS a neighbourhood of infinity in a controlled way; an essential singularity avoids nothing. It is also a gateway: it is the elementary cousin of the far deeper Picard theorem, which replaces 'comes arbitrarily close to every value' with the staggering 'actually attains every value, with at most one exception.' Casorati-Weierstrass is what you can prove with bare hands; Picard needs heavy machinery.
For f(z) = e^(1/z) near 0: want a value close to 5? Solve e^(1/z) = 5, i.e. 1/z = ln 5 + 2 pi i n, giving z = 1/(ln 5 + 2 pi i n). For large n these z are as close to 0 as you like, yet f hits exactly 5 there — and the same works for any target except 0.
Inside any disk around 0, e^(1/z) attains values arbitrarily close to (here exactly) every target.
Casorati-Weierstrass says the values come arbitrarily CLOSE to every w, not that every w is hit — that stronger statement (every value bar at most one is actually attained) is Picard's theorem, which is genuinely deeper.