# Questions tagged [analysis]

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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### Preimage of multivariable function is convex

Let $\phi:\mathbb R^n\to \mathbb R\in C^2(\mathbb{R^n})$, such that $D^2\phi$ is positive semi-definite and let $c \in\mathbb{R}$ be arbitrary. Show that $\phi^{-1}((-\infty,c])$ is convex. Hint: ...
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### Lebesgue measurable set for a function

Let $E⊂[0,∞).$ Show the following for the function $f(x)=x^{2}$: The requirement and sufficient condition for $m^*(E)=0$ is $m^*(f(E))=0.$ (Guiding: $E_n=E∩(0,n)$ use the set of sets.) Depicts the ...
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### Bounded analytic in a region is constant

Let $f:G\to\mathbb{C}$ is an analytic function where $G$ is a region in $\mathbb{C}$ such that $|f(z)|\leq|f(\alpha)|$ for some $\alpha\in G$. Then $f$ is a constant function. Can somebody help how ...
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### Prove that there exists a unique $x^* \in X$ such that $T(x^* ) = x^*$ . [closed]
Suppose $(X, \rho)$ is a complete metric space, and suppose the function $T : (X, ρ) \rightarrow (X, ρ)$ is such that $T_n = T ◦ T ◦ · · · ◦ T$ (n times) is a contraction map for some $n \ge 2$. Prove ...