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 Aug 24 comment Proving pointwise convergence to a “Dirichlet-like” function I am not sure what you mean. Can you elaborate further on your hint? I thought I had answered your questions. Aug 23 comment Proving pointwise convergence to a “Dirichlet-like” function Building off of this, would I suppose to the contrary that there is a sufficiently large $N_0$ so that there is an $x \in [0,1]$ for which $f_{N_0}(x) = 0, f(x) = 1$. Taking $n \ge N_0$, I have that for all $\epsilon > 0$, $|f_n(x) - f(x)| \le \epsilon. This can be done as many times as necessary to achieve arbitrary precision. Is this the lines along which you would approach? Aug 23 comment Proving pointwise convergence to a “Dirichlet-like” function Do I appeal to the countability of$\Bbb Q$and show that$|\Bbb Q| = |\Bbb N|$implies pointwise convergence? Aug 23 asked Proving pointwise convergence to a “Dirichlet-like” function Aug 23 accepted Sequence of$C^1[0,1]$functions$(f_n) \to f$but$f \notin C^1[0,1]$Aug 23 comment Sequence of$C^1[0,1]$functions$(f_n) \to f$but$f \notin C^1[0,1]$At least that's what I believe I'm looking for. The "all" would be replaced by "any" if the question was looking for nowhere differentiable right? Aug 23 comment Sequence of$C^1[0,1]$functions$(f_n) \to f$but$f \notin C^1[0,1]$@Ahriman I'm looking for at least one point. Aug 23 comment Sequence of$C^1[0,1]$functions$(f_n) \to f$but$f \notin C^1[0,1]$@timur I think that is what I mean by "cusps". For example,$|x|$has a cusp at$x = 0$. Aug 23 asked Sequence of$C^1[0,1]$functions$(f_n) \to f$but$f \notin C^1[0,1]$Aug 23 suggested rejected edit on Construct a bijection from$\mathbb{R}$to$\mathbb{R}\setminus S$, where$S$is countable Aug 23 suggested rejected edit on Construct a bijection from$\mathbb{R}$to$\mathbb{R}\setminus S$, where$S$is countable Aug 23 awarded Excavator Aug 23 revised Calculate the remainder when there are division cleaned up substantially Aug 23 suggested approved edit on Calculate the remainder when there are division Aug 23 revised Finding$\lim_{x \to \infty} \left[ {x^{x+1} \over (x+1)^x} - { (x-1)^x\over x^{x-1}}\right]$improved formatting Aug 23 revised Why do complex functions have a finite radius of convergence? OCD LaTeXing Aug 23 suggested approved edit on Why do complex functions have a finite radius of convergence? Aug 23 suggested approved edit on Finding$\lim_{x \to \infty} \left[ {x^{x+1} \over (x+1)^x} - { (x-1)^x\over x^{x-1}}\right]$Aug 23 revised Is there a name for$[0,1]$? OCD LaTeXing Aug 23 suggested approved edit on Is there a name for$[0,1]\$?