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One small step
Weird fact of the day: . Why is it weird? Because the sum of two clearly nonconstant functions is constant. Well, more or less so. How to prove it? Differentiate. How can I use it? You can confuse people with … Continue reading
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Another puzzle thing
This one is going to be more tricky. Imagine a sphere with two linked handles. Is it homeomorphic to a sphere with two unlinked handles? Well, you know the classification theorem, so you know that the answer is “yes”. But … Continue reading
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A nifty trick I learned yesterday: suppose you have a big matrix with integer elements, like this one: and you want to prove that it’s invertible. What do you do? You consider the same matrix modulo 2: now you can … Continue reading
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Triskaidekaphobia And You
Today Prof. K. told me that he considered 13 an unlucky number. He sounded only half joking. “You’re a mathematician, for — sake, how can you say such nonsense!” – I wanted to shout at him, but instead I just … 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
The Grandfather Of All Fractals And The Spikes Of Doom
This post is about the monster of the monsters, Riemann’s continuous almost nowhere differentiable function: Proving that it is continuous is simple enough; proving that it doesn’t have the derivative almost everywhere is more complicated. (It is differentiable on a … Continue reading
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