Tag Archives: counterexample

As you know, there exists a bijection . Cantor demonstrated that. You even know how to construct this bijection: if and their decimal expansions are and then . It may be slightly more complicated than that, because in order to … Continue reading

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Fun with restrictions

If is a function, let us call a restriction of to . Let’s analyze one nice and pretty surprising fact concerning restrictions. Fact: Suppose we have a function of two variables and it’s restriction to every straight line going through … Continue reading

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Saddles, who needs them?

It is not difficult to prove that every continuously differentiable function : Must have a minumum point between two maximums and – If it has precisely one critical point, be it a local minimum or a local maximum, that point must also be a … Continue reading

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Warming Up: Waves, Waves As Far As An Eye Can See

Here is a funny function that I could’t wrap my mind around as a freshman: if , and  . It is both continuous and differentiable, the derivative at zero is 1, because: but at the same time – and that is … Continue reading

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