query: Show that if $f'(z)=0$ in a connected open set in $\mathbb{C}$, then $f(z)=a \in \mathbb{C}$
positive_passages.docid: 154742
positive_passages.text: If $f(z)$ analytic in the domain $D$ and $f'(z)=0$ so $f(z)=$constant?
negative_passages.docid: 62911
negative_passages.text: Let $(X,\tau)$ be a connected topological space, $Y\subset X$ connected and $U\subset X\setminus Y$ clopen. Prove that $ Y \cup U$ is connected
negative_passages.docid: 45720
negative_passages.text: Prove that $∂A$ is closed given $∂A = \text{Cl}(A) − \text{Int}(A)$
negative_passages.docid: 143105
negative_passages.text: Show that, for every real number a, the set {$x ∈ S | f(x) ≤ a$} is a closed subset of $\mathbb R^d$
negative_passages.docid: 32723
negative_passages.text: Open sets and holomorphic functions in Complex analysis
negative_passages.docid: 6722
negative_passages.text: to show $f$ is a polynomial in $z$
negative_passages.docid: 151958
negative_passages.text: How to show $f'(0)$ exist and is equal to $1$?
negative_passages.docid: 87723
negative_passages.text: Show a holomorphic function on unit disc with with $|f'(z) - f'(0)| < |f'(0)|$ is injective
negative_passages.docid: 61640
negative_passages.text: Prove c exists for f:[0,1] -> R
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: 48592
negative_passages.text: Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists
negative_passages.docid: 30204
negative_passages.text: Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
negative_passages.docid: 113892
negative_passages.text: Show $A \in \cal F$ exists, such that $\mathbb P(A)=1$ and $\mathbb Q(A)=0$.
negative_passages.docid: 138026
negative_passages.text: To show that $n!f(x) \in \mathbb{Z}[x]$
negative_passages.docid: 10727
negative_passages.text: Let $f$ be an entire function and $L$ a line in $\mathbb{C}$ such that $f(\mathbb{C})\cap L=\emptyset$. Show that $f$ is constant function.
negative_passages.docid: 100765
negative_passages.text: If $f'(x) = 0$ for all $x \in \mathbb{Q}$, is $f$ constant?
negative_passages.docid: 103125
negative_passages.text: Show that a non-constant holomorphic function has zeroes in a compact set