query: Let $X \subset \mathbb{R^m}$. Show that if any continuous function of real value and defined in $X$ reaches a value maximum, then $X$ is compact.
positive_passages.docid: 51853
positive_passages.text: $K\subseteq \mathbb{R}^n$ is a compact space iff every continuous function in $K$ is bounded.
negative_passages.docid: 155923
negative_passages.text: continuity of $f(x,y)$ in $\mathbb R^2$.
negative_passages.docid: 102143
negative_passages.text: $f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $\int_{0}^{\infty}f(x)dx$ exists.
negative_passages.docid: 143618
negative_passages.text: If a function $f$ is continuous $\implies$ $|f|$ is continuous.(Answered by Myself)
negative_passages.docid: 116542
negative_passages.text: Which of the following sets are compact in $\mathbb{M}_n(\mathbb{R})$[NBHM_PhD Screening Test-2013, Topology]
negative_passages.docid: 64694
negative_passages.text: $C_{c}(X)$ is complete. then implies that $X$ is compact.
negative_passages.docid: 44402
negative_passages.text: If $\{x:f(x) > a\}$ and $\{x:f(x) < a\}$ are open$\forall~a\in\mathbb{R}$ then f is continuous.
negative_passages.docid: 154247
negative_passages.text: Proving $f(x)$ attains $\max$ or $\min$ when $f(x)\to0$ as $|x|\to\infty$.
negative_passages.docid: 46101
negative_passages.text: If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?
negative_passages.docid: 99179
negative_passages.text: Continuous function with compact domain has continuous inverse
negative_passages.docid: 80643
negative_passages.text: Show that f(x) defined on [0,infinity) R to is continuous at 0.
negative_passages.docid: 143105
negative_passages.text: Show that, for every real number a, the set {$x ∈ S | f(x) ≤ a$} is a closed subset of $\mathbb R^d$
negative_passages.docid: 22464
negative_passages.text: Which of the following subsets of   $\mathbb{R}^2$ are compact
negative_passages.docid: 117502
negative_passages.text: Proving that a certain continuous function is surjective.
negative_passages.docid: 120947
negative_passages.text: continuous real-valued function
negative_passages.docid: 18596
negative_passages.text: A set $A \subset l_1$ is compact if and only if closed, bounded, and one other condition
negative_passages.docid: 39076
negative_passages.text: Contraction of compact sets