query: Show that $f(x) = 0$ for all $x \in \mathbb{R}$
positive_passages.docid: 2321
positive_passages.text: Can there be two distinct, continuous functions that are equal at all rationals?
negative_passages.docid: 112285
negative_passages.text: Show that $\lim_{x\to\infty} f(x)=0 \implies \lim_{x\to\infty} f'(x)=0$
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: 2072
negative_passages.text: Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$
negative_passages.docid: 17926
negative_passages.text: Zero divisor in $R[x]$
negative_passages.docid: 110395
negative_passages.text: Integrable function $f$ on $\mathbb R$ does not imply that limit $f(x)$ is zero
negative_passages.docid: 90369
negative_passages.text: Does $\int_a^bf(x) = 0$ imply $f(x) = 0$ for a Riemann Integral function?
negative_passages.docid: 131793
negative_passages.text: Zero integral of measurable $f$ on every interval implies $f=0$?
negative_passages.docid: 130322
negative_passages.text: Show that $f(x) = 0$ if $x = 0$ $f(x) = \text{exp}(-1/x^{2})$ if $x \neq 0$ is infinitely differentiable but not analytic at $x_{0} = 0$ `
negative_passages.docid: 145573
negative_passages.text: How to show that $f$ is a zero function?
negative_passages.docid: 44129
negative_passages.text: If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show that $\lim _{x\to \infty } (f(x+1)-f(x))=0$.
negative_passages.docid: 63418
negative_passages.text: Show that $f_n(x) = \frac{x^2}{n}$ for $x \in \mathbb{R}$ does not converge uniformly towards $0$ for $n \rightarrow \infty$
negative_passages.docid: 48314
negative_passages.text: Show that if a function is not negative and its integral is $0$ than the function is $0$
negative_passages.docid: 76292
negative_passages.text: Showing that $f$ is identically $0$ on $\mathbb{D}$
negative_passages.docid: 66451
negative_passages.text: In a ring, how do we prove that a * 0 = 0?
negative_passages.docid: 95469
negative_passages.text: Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$
negative_passages.docid: 28949
negative_passages.text: Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$