query: If $f\colon\mathbb R\to\mathbb R$ is continuous and satisfies $f(x+y) = f(x) + f(y)$ and $f(1)=1$, then $f(x)=x$ for all $x$.
positive_passages.docid: 152841
positive_passages.text: I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive.
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: 154521
negative_passages.text: Linear functional $f$ is continuous at $x_0=0$ if and only if $f$ is continuous $\forall x\in X$?
negative_passages.docid: 86472
negative_passages.text: $f:\mathbb R \to \mathbb R^n$ be such that $G(f):=\{(x,f(x)):x \in \mathbb R\}$ is closed and connected in $\mathbb R^{n+1}$ , is $f$ continuous ?
negative_passages.docid: 124444
negative_passages.text: If $f$ s discontinuous at $x_0$ but $g$ is continuous there, then $f+g$?
negative_passages.docid: 2093
negative_passages.text: If $f$ is continuous then $G$ is homeomorphic to $X$.
negative_passages.docid: 144614
negative_passages.text: If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .
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: 105652
negative_passages.text: Suppose $f: \mathbb{R} \to \mathbb{R}$ is function such that $f^3$ is continuous on $\mathbb{R}$. Prove that $f$ is continuous on $\mathbb{R}$
negative_passages.docid: 151427
negative_passages.text: f, g continuous for all rationals follow by continuous on all reals?
negative_passages.docid: 130023
negative_passages.text: Prove that the sum of two continuous functions is continuous
negative_passages.docid: 83326
negative_passages.text: $f:\mathbb R\to\mathbb R$ continuous function. Which of the following sets can not be image of $(0,1]$ under $f$?
negative_passages.docid: 74950
negative_passages.text: Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
negative_passages.docid: 82597
negative_passages.text: How to find $ f(x)$ if $f(1-f(x))=x$ for all $x$ $\in \mathbb{R}$
negative_passages.docid: 55605
negative_passages.text: If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
negative_passages.docid: 92089
negative_passages.text: If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$
negative_passages.docid: 143800
negative_passages.text: continuous map on $\mathbb{R}$ which is the identity on $\mathbb{Q}$ is the identity map, hence Aut$(\mathbb{R}/\mathbb{Q})= 1.$