query: Proving $g(x)$ is not a rational function
positive_passages.docid: 151439
positive_passages.text: $\sqrt{1-x^2}$ is not a rational function
negative_passages.docid: 81921
negative_passages.text: If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$
negative_passages.docid: 25050
negative_passages.text: To prove that $(F,+)$ and $(F-\{0\},\cdot)$ are not isomorphic as groups.
negative_passages.docid: 3296
negative_passages.text: Proof That $\mathbb{R} \setminus \mathbb{Q}$ Is Not an $F_{\sigma}$ Set
negative_passages.docid: 27775
negative_passages.text: Proving a function is not a contraction
negative_passages.docid: 35837
negative_passages.text: How to prove that any (integer)$^{1/n}$ that isn't an integer, is irrational?
negative_passages.docid: 67234
negative_passages.text: The function $G: x \mapsto 2^{x^2}$ maps $\mathbb{R}$ onto $\{ x \in \mathbb{R} : x \geq 1 \}$
negative_passages.docid: 43956
negative_passages.text: Differentiating $f(x)^{g(x)}$
negative_passages.docid: 61486
negative_passages.text: Show that $f$ is not differentiable at $(0,0)$ - $\frac{x_1^2x_2}{x_1^2+x_2^2}$
negative_passages.docid: 142870
negative_passages.text: Prove the function is not differentiable at (0,0)
negative_passages.docid: 119545
negative_passages.text: Nonzero rationals under multiplication are not a cyclic group
negative_passages.docid: 90055
negative_passages.text: Show that $\mathcal P(x)\notin x$
negative_passages.docid: 72694
negative_passages.text: Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
negative_passages.docid: 94190
negative_passages.text: Disproving the statement "If $f(z)$ is not an entire function, then $g(z) = f^2(z)$ is not entire.”
negative_passages.docid: 86015
negative_passages.text: $\mathbb Q[x]/(x^2+1)$ is not isomorphic to $\mathbb Q[x]/(x^2+2)$
negative_passages.docid: 132386
negative_passages.text: Show that $(\mathbb{Q}^*,\cdot)$ and $(\mathbb{R}^*,\cdot)$ aren't cyclic
negative_passages.docid: 85092
negative_passages.text: Proving $ f(x) = x^2 $ is not uniformly continuous on the real line