query: Show that if $f$ and $g$ are uniformly continuous on $A\subseteq\mathbb{R}$, then $f + g$ is uniformly continuous on $A$.
positive_passages.docid: 69122
positive_passages.text: If $f,g$ are uniformly continuous prove $f+g$ is uniformly continuous but $fg$ and $\dfrac{f}{g}$ are not
negative_passages.docid: 93461
negative_passages.text: Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty.$Prove that $f$ is uniformly continuous.
negative_passages.docid: 62723
negative_passages.text: If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere
negative_passages.docid: 109570
negative_passages.text: How to show that the set of points of continuity is a $G_{\delta}$
negative_passages.docid: 131060
negative_passages.text: Is $\sqrt{x}$ uniformly continuous in $\mathbb{R}^+$?
negative_passages.docid: 7876
negative_passages.text: If $f(x+y)=f(x)+f(y) ,\forall\;x,y\in\Bbb{R}$, then if $f$ is continuous at $0$, then it is continuous on $\Bbb{R}.$
negative_passages.docid: 12408
negative_passages.text: Prove that a function whose derivative is bounded is uniformly continuous.
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: 83494
negative_passages.text: Uniform continuity of $\ln(x)$
negative_passages.docid: 60282
negative_passages.text: Prove that $f_n$ converges uniformly on $\mathbb{R}$
negative_passages.docid: 46629
negative_passages.text: Prove if $f(x) = g(x)$ for each rational number x and $f$ and $g$ are continuous, then $f = g$
negative_passages.docid: 57023
negative_passages.text: If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$
negative_passages.docid: 143695
negative_passages.text: Is an increasing, bounded and continuous function on $[a,+\infty)$ uniformly continuous?
negative_passages.docid: 14067
negative_passages.text: If f uniformly continous then $|f(x)|\leq C\cdot (1+|x|)$ for constant C
negative_passages.docid: 9614
negative_passages.text: How does the existence of a limit imply that a function is uniformly continuous
negative_passages.docid: 40915
negative_passages.text: Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so f bounded.
negative_passages.docid: 130023
negative_passages.text: Prove that the sum of two continuous functions is continuous