Continuous Memes

Ever notice how continuous functions become discrete approximations when you're trying to integrate in the bathtub? Left side: your elegant double integral in all its continuous glory. Right side: the same cat broken down into a finite sum of tiny cat chunks. Mathematicians call this "numerical integration," cats call it "existential crisis." Next time your calculus professor talks about approximating areas, just remember this feline's journey from continuous to discrete—it's literally the perfect visual proof that everything can be broken down into smaller pieces. Even dignity.

The mathematical glow-up we never knew we needed! On the left, Sigma (Σ) is represented by Porygon, a Pokémon made of sharp polygons and straight edges—perfect for discrete summation. On the right, the integral symbol (∫) gets its curvy representation through Porygon's evolved form, Porygon2, all smooth and rounded. It's literally the transition from calculating discrete sums to continuous integration, but with pocket monsters. Math professors should start using this in Calculus 101 immediately.

First, our cheerful math student understands the well-behaved parabola. Next, she comprehends functions with occasional differentiability hiccups. But then... Weierstrass function has entered the chat . That last panel shows the mathematical equivalent of Lovecraftian horror—a continuous function that's differentiable precisely nowhere . It's like nature decided "Let's create something that follows exactly one rule while breaking every intuition humans have about smoothness." Every calculus professor has witnessed this exact facial expression when introducing pathological functions. The math gods created these monstrosities specifically to crush undergraduate spirits and keep mathematicians humble.

Perfect visual representation of mathematical operations in 3D modeling. The summation symbol (Σ) represents discrete, chunky polygonal rendering—basically counting individual faces. Meanwhile, the integral symbol (∫) gives us those smooth, continuous curves that make calculus professors swoon. It's the difference between counting stairs and sliding down a ramp. Next time someone asks why calculus matters, just point to their unnaturally smooth video game character.

The staircase perfectly captures the fundamental difference between summation and integration. The left side with discrete steps represents summation (Σ) - counting each individual step as you climb. Meanwhile, the ramp on the right represents integration (∫) - a smooth, continuous path covering the same vertical distance. Mathematicians will tell you they're equivalent in the limit, but engineers know which one is wheelchair accessible. The real question: would calculus have been invented sooner if Newton had mobility issues?