query: Suppose $f : \space [-1,1] \subset \mathbb{R} \rightarrow \mathbb{R}$
positive_passages.docid: 157608
positive_passages.text: One-sided limits of $f'(x)$ at a point vs. one-sided limits of the difference quotient at that point
negative_passages.docid: 158498
negative_passages.text: $A\subset \mathbb{R}$ with more than one element and $A/ \{a\}$ is compact for a fixed $a\in A$
negative_passages.docid: 153869
negative_passages.text: Under what $p$, $-1$ is a square in $\mathbf F_p$?
negative_passages.docid: 14975
negative_passages.text: Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$
negative_passages.docid: 40209
negative_passages.text: $f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
negative_passages.docid: 117716
negative_passages.text: Condition on function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $(a,b)\mapsto | f(a) - f(b)|$ generates a metric on $\mathbb{R}$
negative_passages.docid: 93102
negative_passages.text: prove that $f:X\rightarrow Y$ is surjective if and only if $f(f^{-1}(C))=C$
negative_passages.docid: 54321
negative_passages.text: Questions about $f: \mathbb{R} \rightarrow \mathbb{R}$ with bounded derivative
negative_passages.docid: 60655
negative_passages.text: $\ker T\subset \ker S\Rightarrow S=rT$ when $S$ and $T$ are linear functionals
negative_passages.docid: 32063
negative_passages.text: if $f'(x)\rightarrow L$ as $ x \rightarrow \infty$, $-\infty \leq L \leq \infty $ then $ f(x)/x \rightarrow L $ as $x \rightarrow \infty$
negative_passages.docid: 66895
negative_passages.text: A condition for the image of a metrizable space to be metrizable
negative_passages.docid: 84143
negative_passages.text: Given any function $g(x): \mathbb{R} \rightarrow \mathbb{R}$, find a function $f(x)$ such that $f(f(x)) = g(x)$.
negative_passages.docid: 118154
negative_passages.text: $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ twice
negative_passages.docid: 80827
negative_passages.text: Let $A$ be a nonempty subset of $\mathbb{R}$. Prove...
negative_passages.docid: 49220
negative_passages.text: Open Sets of $\mathbb{R}^1$ and axiom of choice
negative_passages.docid: 30860
negative_passages.text: Prove that $f(x)=0$ in $[-1,1]$.
negative_passages.docid: 17212
negative_passages.text: The preimage of a subset