query: Showing that $x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = 0$ implies $f$ is constant.
positive_passages.docid: 153191
positive_passages.text: A function satisfy $x\frac{\partial f(x,y)}{\partial x}+y\frac{\partial f(x,y)}{\partial y}=0$ in a convex domain implies it is a constant
negative_passages.docid: 61190
negative_passages.text: If all derivatives of $f$ are uniformly bounded by a common constant and $f(1/n) = 0$ for all $n$, is $f$ identically zero?
negative_passages.docid: 46310
negative_passages.text: Showing that $ \int_{0}^{1} \frac{x-1}{\ln(x)} \mathrm dx=\ln2 $
negative_passages.docid: 26388
negative_passages.text: proof that function is constant
negative_passages.docid: 106290
negative_passages.text: Prove a function is constant. (complex analysis)
negative_passages.docid: 136782
negative_passages.text: Prove that $x * y = \frac{x+y}{1+xy}$ is a stable part of $G=(-1, 1)$
negative_passages.docid: 49408
negative_passages.text: Proof of the derivative of ln(x)
negative_passages.docid: 53767
negative_passages.text: How to prove that derivatives have the Intermediate Value Property
negative_passages.docid: 77133
negative_passages.text: $g'(x) = \frac{1}{x}$ for all $x > 0$ and $g(1) = 0$. Prove that $g(xy) = g(x) + g(y)$ for all $x, y > 0$.
negative_passages.docid: 143491
negative_passages.text: Show that $f$ is differentiable on $(−1, 1)$.
negative_passages.docid: 104510
negative_passages.text: For continuous $f$, $\int_0^x f(t)dt = ∫_x^1f(t)dt$ implies $f=0$.
negative_passages.docid: 142138
negative_passages.text: Show that $f(x)$ is a constant function if $\lim\limits_{h\to 0}\frac{1}{h^3}\int_{-h}^{h}f(x+t)\cdot t\,dt=0$ for all $x$
negative_passages.docid: 11158
negative_passages.text: Proof that a continuous function with $f(x) = f(x^2)$ is constant.
negative_passages.docid: 107727
negative_passages.text: $m\{x\in [0,1]:f'(x)=0\}>0$
negative_passages.docid: 23615
negative_passages.text: Why does zero derivative imply a function is locally constant?
negative_passages.docid: 158407
negative_passages.text: derivative bounded by a constant multiple of the function
negative_passages.docid: 76292
negative_passages.text: Showing that $f$ is identically $0$ on $\mathbb{D}$