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Let ##f:ℝ→ℝ## such that f is piecewise linear, which means, for every ##x \in ℝ##, there is an ##ε>0## such that f restricted to ##[x-ε,x]## and restricted to ##[x,x+ε]## are linear functions. Find the cardinality of ##A##={##f:ℝ→ℝ## / ##f## is piecewise linear}

The attempt at a solution.

I couldn't get further than: ##|A|##≤|{##f:ℝ→ℝ##}=##c^c## but this doesn't get me anywhere. I know that not all linear piecewise functions are continuous, so I cannot assume that ##|A|≤c##. I'm trying to think of a bijective function between A and another set which I know the cardinality of. Maybe the set A is a subset of something with cardinality c, but I'm not sure. If this was true, then I could find an injective function from a set B of cardinality c to A to conclude ##c=|B|≤|A|≤c## ##→## ##|A|##=##c##.