Again, there are very sudden changes in value that you have to hold in your head. Until the limit gets all the way to infinity, there is always a slope to the function as it passes through its transition.Īnother similar place this puzzle arrises in signal processing where we have the concept of the Dirac Delta Function ( ). Each individual sine wave component is a proper function that passes the Vertical Line Test. Another way to compose a square wave is as an infinite sum of sine waves (as in this Fourier Series video sequence). Wolfram lists three ways to analytically represent a square wave using functions. We need to exercise some flexibility in order to resolve the conflict. It is a good example of our quest for beautiful tidy mathematical rigor bumping up against something that is super useful in real life, in this case, the highly non-linear square wave.
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