Suppose we have a sequence of continuous functions f0, f1, f2, …, and they happen to converge pointwise to a limit function fn(x) → f(x). Must the limit function also be continuous? Cauchy made a mistake about this, claiming that a convergent series of continuous functions is continuous. But this turns out to be incorrect. For a counterexample, consider the functions x, x2, x3, …on the unit interval, as pictured here in blue. These functions are each continuous, individually, but as the exponent grows, they become increasingly flat on most of the interval, spiking to 1 at the right. The limit
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