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Interactive Explainers

Play with the theorems

Interactive explainers that let you move the mathematics: drag ε and watch δ answer, walk the topologist's sine curve, trap a subsequence by bisection, watch Newton's method fall into a cycle. 15 are free to play right here — the rest come with Trap Analysis.

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Video explainers

Narrated walk-throughs of the trickiest ideas, made for this site — no ads, no recommendations pulling you elsewhere. Free to watch whether or not the interactive below is unlocked.

Unit 1Continuity and Differentiation › Continuity, uniform continuity, Lipschitz#

The ε–δ limit machineinteractivefree

Unit 1Sequences and Series of Functions › Pointwise vs uniform convergence, M-test, Dini#

Watch uniform convergence fail: xⁿ on [0, 1]interactivefree

Unit 1Integration › Riemann integration and criteria#

Riemann sums closing on the integralinteractivefree

Unit 1Metric Spaces › Open/closed sets, limit points, closure, interior#

The unit ball, from diamond to squareinteractivefree

Unit 1Functions of Several Variables › Partial derivatives, differentiability, chain rule#

Partial derivatives can exist while the function is not differentiableinteractivefree

Step 1 / 4The candidate

Let

f(x,y)=xyx2+y2 for (x,y)≠(0,0),f(0,0)=0.f(x,y) = \frac{xy}{x^2+y^2} \text{ for } (x,y) \neq (0,0), \qquad f(0,0)=0.

Is ff differentiable at the origin?

Unit 1Metric Spaces › Compactness: open covers, sequential, Heine–Borel#

Why (0,1) is not compact — an explicit cover with no finite subcoverinteractivefree

Step 1 / 4The claim

(0,1)(0,1) is bounded, so if compactness were only about boundedness it would qualify. Show directly that it is not compact by exhibiting an open cover with no finite subcover.

Unlock every visual

Every interactive explainer comes with Trap Analysis — along with the worked solutions, the per-option trap diagnosis and the counterexample objects. The papers, the book and the video map are free without it.

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