Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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Help with a DCT problem

The question is Let ${p_t}\left( x \right) = \frac{1}{{\sqrt {2\pi t} }}{e^{ - \frac{{{x^2}}}{{2t}}}},t > 0,x \in \Bbb{R}$. It is known that $\int_R {\frac{1}{{\sqrt {2\pi t} }}{e^{ - ...
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Solving ODE rigorously

I am given the ODE $$(f''(r)+\frac{f'(r)}{r})(1+f'(r)^2)-f'(r)^2f''(r)=0$$ and want to solve it rigorously for $r>0.$ So especially, I don't want to loose any solutions. $\textbf{Derivation of ...
2
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How much does convolution with a C^m kernel increase the order of continuity.

EDIT: I rewrote the question, see the edit history for my previous attempt. My questions are: Is the following proof/reasoning essentially correct? How do I make it more precise, concise, correct? ...