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Weierstrass extreme value theorem

Level UniversityDifficulty ★★★★★Theorem⌖ Open in the map

What is it?

A continuous function on a closed bounded interval attains a maximum and a minimum. Without it, "find the best solution" might have no answer.

Statement

$f \in C[a,b] \implies \exists x_m, x_M \in [a,b]:\ f(x_m) \le f(x) \le f(x_M)\ \forall x \in [a,b]$ .

Idea of the proof

Take $x_n$ with $f(x_n) \to \sup f$ ; Bolzano–Weierstrass gives a convergent subsequence $x_{n_k} \to c$ , and continuity gives $f(c) = \sup f$ .

Where it shows up in computing

Operations research and logistics★★★★★advancedOptimization and systems
Existence of an optimal solution: a continuous cost on a compact feasible set always has a minimum.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

ℒ AI and machine learning

Rolle's theorem→Mean value theorem→Lipschitz continuity→Gradient descent★★★★★
Rolle's theorem→Mean value theorem→Lipschitz continuity→Gradient descent→Learning rate★★★★★
Rolle's theorem→Mean value theorem→Lipschitz continuity→Gradient descent→Backpropagation★★★★★
Rolle's theorem→Mean value theorem→Lipschitz continuity→Gradient descent→Stochastic gradient descent (SGD)★★★★★
Rolle's theorem→Mean value theorem→Lipschitz continuity→Gradient descent→Loss landscape★★★★★
Rolle's theorem→Mean value theorem→Lipschitz continuity→Gradient descent→Reinforcement learning★★★★★
+3

⇄ Optimization and systems

Rolle's theorem→Mean value theorem→Lipschitz continuity→Gradient descent→Recommender systems★★★★★
Rolle's theorem→Mean value theorem→Fundamental theorem of calculus→Cumulative distribution function→Queueing theory and performance★★★★★
Operations research and logistics★★★★★

⚛ Physics and simulation

Rolle's theorem→Mean value theorem→Fundamental theorem of calculus→Physics engines★★★★★
Rolle's theorem→Mean value theorem→Fundamental theorem of calculus→Physics engines→N-body gravitational simulation★★★★★
Rolle's theorem→Mean value theorem→Fundamental theorem of calculus→Physics engines→Fluid dynamics and CFD★★★★★

ψ Quantum computing and physics

Rolle's theorem→Mean value theorem→Taylor's theorem and the remainder→Taylor and Maclaurin series→Quantum computing★★★★★

What depends on it

Rolle's theorem