query: How to prove that $f(f(x))=-x$ implies that $f$ is not continuous?
positive_passages.docid: 11453
positive_passages.text: Find a real function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?
negative_passages.docid: 93021
negative_passages.text: $f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$
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 $f(c) = c$?
negative_passages.docid: 145017
negative_passages.text: Example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points
negative_passages.docid: 141223
negative_passages.text: $f(x^2) = 2f(x)$ and $f(x)$ continuous
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: 59063
negative_passages.text: $f(f(x))=f(x)$ question
negative_passages.docid: 69869
negative_passages.text: if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
negative_passages.docid: 153730
negative_passages.text: Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.
negative_passages.docid: 113687
negative_passages.text: $f$ is continuous function from $ [0,1]$ to itself and f(0)=0, f(1)=1 and $f(f(x))=x$ then show that $f(x)=x$ on$[0,1]$
negative_passages.docid: 20273
negative_passages.text: Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.
negative_passages.docid: 47381
negative_passages.text: If $\lim_{x\to\infty} [f(x+1)-f(x)] =l$ then $\lim_{ x\to\infty}f(x)/x =l$ ($f$ is continuous)
negative_passages.docid: 158888
negative_passages.text: Prove that the function $f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$.
negative_passages.docid: 136011
negative_passages.text: Must $f$ be continuous?
negative_passages.docid: 110146
negative_passages.text: I need to prove the continuity of $f(x)=\log x$ using a $\epsilon-\delta$ proof
negative_passages.docid: 150554
negative_passages.text: proof that $f'$ is continuous at $a$
negative_passages.docid: 144013
negative_passages.text: If $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$.