query: Show that $A$ defined by a continuous function is open
positive_passages.docid: 143105
positive_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: 127900
negative_passages.text: Interior of closure of open set in topological space
negative_passages.docid: 130975
negative_passages.text: Given a pair of continuous functions from a topological space to an ordered set, how to prove that this set is closed?
negative_passages.docid: 126461
negative_passages.text: Show that $g:I\to\mathbb{R}$ is continuous and $f:(x-x_0)g(x)$ is differentiable
negative_passages.docid: 65346
negative_passages.text: Characterisation of uniformly continuous function
negative_passages.docid: 118332
negative_passages.text: proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
negative_passages.docid: 158423
negative_passages.text: How to prove that if $f$ is continuous a.e., then it is measurable.
negative_passages.docid: 151573
negative_passages.text: Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
negative_passages.docid: 14896
negative_passages.text: Prove: the set of zeros of a continuous function is closed.
negative_passages.docid: 94781
negative_passages.text: Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
negative_passages.docid: 122086
negative_passages.text: continuous function: preimage of closure of a set is closure of preimage
negative_passages.docid: 75735
negative_passages.text: Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$)
negative_passages.docid: 63701
negative_passages.text: An example of a regular function over an open set
negative_passages.docid: 12523
negative_passages.text: Supremum of a Continuous Function is Continuous
negative_passages.docid: 74364
negative_passages.text: Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$
negative_passages.docid: 66531
negative_passages.text: $\|h(x)-h(y)\|\geq 3\|x-y\|$ prove that image of open is open
negative_passages.docid: 132013
negative_passages.text: How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?