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Let $X$ be a Baire space, and $(Y,d)$ a metric space. Assume that $f_n :X \rightarrow Y$ are continuous functions that pointwise convergence to $f :X \rightarrow Y$.

a) Prove that $A=\{x\in X | f$ is continuous on $x \}$ is dense in $X$.

b) Prove that if $X=Y=\mathbb{R}$, then $A$ isn't countable.

Any ideas?

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    See https://www.math.ucsd.edu/programs/undergraduate/1213_honors_presentations/Siuyung_Fung_Honors_Thesis.pdf for a) – 2017-01-07
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    @HennoBrandsma I'm sorry, I couldn;t find it there...Which Thm is it? – 2017-01-07
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    2.2 and its corollary 2.3. Note that $f$ is a Baire class 1 function by definition 2.1. – 2017-01-07
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    b) then follows easily from this as well. – 2017-01-07
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    @HennoBrandsma I understood a). For b),$A$ isn't countable because it's dense in $\mathbb{R}$, right? – 2017-01-07
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    Dense sets can be countable (e.g. the rationals). But a second category set in the reals cannot be. – 2017-01-07
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    @HennoBrandsma Because if A second categroy set could be countable,we could see it as a union of countable singletons which makes it from the first category? – 2017-01-07
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    That's indeed a correct argument for that case. – 2017-01-07

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