Your comment has a minus sign but your answer doesn't... did you intend them to match?

@StevenGubkin: That's not really better considering I don't really see people doing 51st-degree Taylor series, for example. The whole point was to get a realistic example. But in this case, I would've cited RKF45 which uses a 5th-order error estimator.

Z-transform isn't an algorithm but rather a function... maybe you meant Chirp-Z transform?

Do you happen to know of any example that isn't constructed piecewise or by an infinite series or something else that students might try to blame it on?

By the way, it seems you forgot to edit your answer to clarify this...

Easy counterexample: $6 = 2 \times 3$ but $6$ does not divide $2$ or $3$...

What a gem! I always thought this was true.

This is so, so, so counterintuitive!! I love it.

@MartinBrandenburg: I don't understand the locally-constant hint. Would you mind giving an actual counterexample?

@PaulF: ionno, but say if my father had 2 houses and promised to give me half of them, I would not expect him to demolish half of each and then leave me the other half of each...

@Sumyrda: I wouldn't call that a camel any more than I'd call my hamburger a cow...

I like how you said this is "hard to do", because, I guess, never claim something obviously impossible is impossible... :P

PS Archimedes link here.

For what it's worth, Archimedes apparently found the area under a parabola. 2300 years ago. I don't know if you call that integration though.

Should probably be on Math.SE rather than here... I also wouldn't really call this "convex analysis"...

@AntonPetrunin: If I understand this example correctly, the subgradient (1,0) can be approximated arbitrarily well by a convex combination of gradients in a ball around (0,0)... so we might as well call it a subgradient, right? If so, it seems a bit like telling someone that $\max x : x < 1$ doesn't exist... which is true, but not all that interesting. Are there any interesting counterexamples, or are they all correct if we're willing to take reasonable limits as needed?