# Tagged Questions

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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### Derivative of matrix inversion function?

Let's say I have a function $f$ which maps any invertible $n\times n$ matrix to its inverse. How do I calculate the derivative of this function?
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### how to prove that $k+1 \ge (1+\frac{1}{k})^{k}$?

How to prove that $$k+1\ge \bigg(1+\frac{1}{k}\bigg)^{k}$$ when $k>2$
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### Change of variable in differential equation legitimate?

Just a general question ( I don't want to solve this ODE, I just want to understand why this is legitimate to do or not): Assuming we have the ODE $$y'(x) - \cos(x) y(x)=0$$ on $[0,2\pi]$ Am I ...
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### Continuous linear functional and weak convergence

I have a question about a continuous linear functional. $T>0$ : fix. $C([0,T]):=\{w:[0,T]\to \mathbb{R}\,;\, w \,{\rm is\,conti.} \}$ $C_{0}([0,T]):=\{w \in C([0,T]) \,; \,w(0)=0 \}$ Then ...
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### Absolute value inequality, difficult proof

Prove that $|x - y| \le |x| + |y|$ Let $x > y$ without the loss of generality, $x - y > 0 \implies |x - y| > 0$ $|x| > 0, |y| > 0 \implies |x| + |y| > 0$ But how can you ...
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### Every infinite subset of E in R having a limit point in E implies E is closed and bounded

Every infinite subset of E in R having a limit point in E implies E is closed and bounded. Could you please help with a formal proof of this result ?
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### Spivak “min” notation confusion

Spivak uses a notation: min$(1, \frac{\epsilon}{2|a| + 1})$ What does he mean by this notation? especially by "min"??
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### How to deduce the following trig relation?

How can I deduce: $$\sqrt{|x|}\sin(\frac{1}{x}) \le \sqrt{|x|}$$?? I know of the relation. $$\sin(u) \le u$$ $$u = \frac{1}{x}$$ $$\sin(1/x) \le \frac{1}{x}$$ But nothing related to $\sqrt{x}$ ...
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### References on integration: collections of fully worked problems (and explanations) of (1) advanced and (2) unusual techniques

I am searching for two kinds of books. (1) Comprehensive books that collect, explain, and provide many examples (that is, fully worked problems) of advanced integration techniques (that is, ...