Counterexamples Memes

The mathematical descent into madness is real! Complex analysis is like that chill friend who makes everything seem elegant—one derivative means infinite differentiability, closed path integrals conveniently equal zero, and bounded entire functions are reassuringly constant. Life is beautiful! Meanwhile, real analysis is that friend who destroys your sanity by introducing counterexamples to everything you thought was true. You start confidently, then discover functions so pathological they can't even be graphed. The Weierstrass function? Continuous everywhere but differentiable nowhere! The Devil's staircase? Differentiable almost everywhere with derivative zero, yet still manages to increase! No wonder mathematicians end up cackling maniacally about undrawable functions.