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I'm having trouble showing that the set of differentiable functions on $[0,1]$ is coanalytic ($\mathbf{\Pi}_{1}^{1}$) and the set of continuously differentiable functions of $[0,1]$ is analytic ($\mathbf{\Sigma }_{1}^{1}$). Any help would be appreciated.

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Hint for showing that $\mathrm{DIFF} := \{f \in C[0,1] | \forall x [f'(x) \, \text{exists}]\}$ is $\mathbf{\Pi}_{1}^{1}$: First show that $\{(f,x)\in C[0,1] \times [0,1]| f'(x) \, \text{exists}\}$ is Borel. This can be done by using the fact that $f$ is continuous and then looking at the differences in secant slopes at rational points. I'll elaborate if you'd like. I suppose the second question should be similar.