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, integration through the fundamental theorem of calculus.

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Conditions on $f:[a,b]\to \mathbb{R}$ to ensure a finite number of crossings

Suppose $f:[a,b]\to \mathbb{R}$ is differentiable except at a finite number of points. Can you give me some examples of additional hypotheses that would guarantee that $$\text{For any $c\in ...
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Question in Munkres Analysis on manifolds

I am trying to solve this problem, but I am having such a hard time trying to solve it. I am Ok with part $(a)$, but parts b and c I dont understand :s
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Biorthogonal functions in $L^p$

I asked one question that is already answered: 1.) I have a question about Lemma 9.5 on page 93/94 reference. It's about the part of the proof where the sequence of $(g_n^*)$ are introduced. I don't ...
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Question about integration on a box

Let $Q \subseteq \mathbb{R}^n$, and $f: Q \to \mathbb{R} $ is integrable over $Q$. $f \geq 0$. if $A \subseteq Q$, then $\int_Q f \geq \int_A f $ Attempt: say $\epsilon > 0$ Let $P_1$ be a ...