query: On the assumption that $f$ is $C^1$, prove that if $D=\{x:f'(x)=0\}$, then measure of $f(D)$ is 0.
positive_passages.docid: 18889
positive_passages.text: Measure of image of critical points set is equal 0
negative_passages.docid: 30860
negative_passages.text: Prove that $f(x)=0$ in $[-1,1]$.
negative_passages.docid: 4656
negative_passages.text: Prove that there is a $\delta$ such that $\int_{0}^{1} (f(x))^2dx\leq \delta$$\int_{0}^{1} (f'(x))^2dx$ for all $f$ with these conditions
negative_passages.docid: 107727
negative_passages.text: $m\{x\in [0,1]:f'(x)=0\}>0$
negative_passages.docid: 55016
negative_passages.text: Prove that if ($d(x,y)=0$ iff $x=y$) and if $d(x,z)\leq d(x,y)+d(z,y)$ then $d$ is a metric
negative_passages.docid: 95260
negative_passages.text: Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e.
negative_passages.docid: 150502
negative_passages.text: If $f(0) = 0$ and $|f'(x)|\leq |f(x)|$ for all $x\in\mathbb{R}$ then $f\equiv 0$
negative_passages.docid: 105068
negative_passages.text: Proving that the function is differentiable and $f'(0)=0$
negative_passages.docid: 49155
negative_passages.text: $f: \Omega \rightarrow \Omega$ holomorphic, $f(0) = 0$, $f'(0) = 1$ implies $f(z) = z$
negative_passages.docid: 32506
negative_passages.text: Prove $kf(x)+f'(x)=0 $ when conditions of Rolle's theorem are satisfied .
negative_passages.docid: 107795
negative_passages.text: If $\,f''(x) \ge f(x)$, for all $x\in[0,\infty),$ and $\,f(0)=f'(0)=1$, then is $\,f(x)>0$?
negative_passages.docid: 18261
negative_passages.text: Prove $\int_{a}^{b}{f(x)dx}>0$.
negative_passages.docid: 111756
negative_passages.text: How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
negative_passages.docid: 1641
negative_passages.text: Proving $f' = - \frac{\partial{F}/\partial{x}}{\partial{F}/\partial{y}}$
negative_passages.docid: 154742
negative_passages.text: If $f(z)$ analytic in the domain $D$ and $f'(z)=0$ so $f(z)=$constant?
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: 75950
negative_passages.text: $C^1$ function on the ball hits $0$