query: Prove $f(x) =0$ for all $x \in [a, b]$
positive_passages.docid: 2321
positive_passages.text: Can there be two distinct, continuous functions that are equal at all rationals?
negative_passages.docid: 97104
negative_passages.text: Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
negative_passages.docid: 6825
negative_passages.text: If $f: \mathbb Q\to \mathbb Q$ is a homomorphism, prove that $f(x)=0$ for all $x\in\mathbb Q$ or $f(x)=x$ for all $x$ in $\mathbb Q$.
negative_passages.docid: 61640
negative_passages.text: Prove c exists for f:[0,1] -> R
negative_passages.docid: 74583
negative_passages.text: Prove that $f'(0)=L$.
negative_passages.docid: 11715
negative_passages.text: Prove that if $\langle Tx,x\rangle =0$ for all $x \in X$, then $T = 0$
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: 57193
negative_passages.text: Prove that $f$ has derivatives of all orders at $x=0$, and that $f^{(n)}(0)=0$ for $n=1,2,3\ldots$
negative_passages.docid: 112323
negative_passages.text: Showing that $f = 0 $ a.e. if for any measurable set $E$, $\int_E f = 0$
negative_passages.docid: 69707
negative_passages.text: If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$.
negative_passages.docid: 60348
negative_passages.text: How do I prove $F(a)=F(a^2)?$
negative_passages.docid: 76610
negative_passages.text: Show that $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$
negative_passages.docid: 88278
negative_passages.text: Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective
negative_passages.docid: 118468
negative_passages.text: $\int f=0$ implies $f=0$ almost everywhere
negative_passages.docid: 88311
negative_passages.text: Linear functional f(v)=0 imply v=0
negative_passages.docid: 138054
negative_passages.text: Proving that $\int_0^1 f(x)e^{nx}\,{\rm d}x = 0$ for all $n\in\mathbb{N}_0$ implies $f(x) = 0$
negative_passages.docid: 28007
negative_passages.text: Assuming: $\forall x \in [0,1]:f(x) > x$ Prove: $\forall x \in [0,1]:f(x) > x + \varepsilon $