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I have to find the cardinality of the set of the non-continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$.

I think we should look for function that at least have a point of discontinuity, but i don't have a clue, really. Any hint?

Michael Hardy
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Ariel Marcelo Pardo
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  • One approach: There are "few" continuous functions, since a continuous function is determined by its values on the rationals. But there are lots of functions. – André Nicolas Jul 14 '15 at 03:41

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You are starting in the wrong direction. If you look at the definition of a function, almost every function is discontinuous. We do a disservice to students when we say "plot a function on the board" and draw something continuous. If you are coming up from below, there is only one value for $f(x)$ that makes the function continuous and continuum many that make it discontinuous. Think about the total number of functions there are, and you can ignore the continuous ones.

Ross Millikan
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