query: Assume that $ f: R \to R $ is uniformly continuous. prove that there are constants A,B such that $ |f(x)| \le A + B|x| $ for all $ x \in R $.
positive_passages.docid: 64955
positive_passages.text: Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$
negative_passages.docid: 12523
negative_passages.text: Supremum of a Continuous Function is Continuous
negative_passages.docid: 48592
negative_passages.text: Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists
negative_passages.docid: 126212
negative_passages.text: Proving uniform continuity on an interval
negative_passages.docid: 8179
negative_passages.text: If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
negative_passages.docid: 131992
negative_passages.text: A problem on continuous functions
negative_passages.docid: 113871
negative_passages.text: Prove $f(x) = \sqrt x$ is uniformly continuous on $[0,\infty)$
negative_passages.docid: 26388
negative_passages.text: proof that function is constant
negative_passages.docid: 9614
negative_passages.text: How does the existence of a limit imply that a function is uniformly 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: 59837
negative_passages.text: Suppose $\mu(X)<\infty$. Show that if $\ f\in\ L^q(X,\mu)$ for some $1\le q<\infty$ then $\ f\in\ L^r(X,\mu)$ for $1\le r<q$
negative_passages.docid: 116056
negative_passages.text: Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$.
negative_passages.docid: 11158
negative_passages.text: Proof that a continuous function with $f(x) = f(x^2)$ is constant.
negative_passages.docid: 57180
negative_passages.text: Techniques to prove a function is uniformly continuous
negative_passages.docid: 13257
negative_passages.text: How to prove this problem about uniform continuity
negative_passages.docid: 41482
negative_passages.text: $f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.
negative_passages.docid: 65346
negative_passages.text: Characterisation of uniformly continuous function