Let $f,g: \mathbb{R}^n → \mathbb{R}^k$ be continuous functions and suppose that $D \subset \mathbb{R}^n$ is a dense set. If $f(x)=g(x)$ for every $x \in D$, then $f(x)=g(x)$ for every $x \in \mathbb{R}^n$.
Asked
Active
Viewed 57 times
-2
-
Possible duplicate of $f,g$ continuous from $X$ to $Y$. if they are agree on a dense set $A$ of $X$ then they agree on $X$ – Tom Collinge May 27 '18 at 11:46
1 Answers
2
Assume that there is a $y$ such that $f(y) \neq g(y)$. Then there exist open sets $U$ and $V$ in $\mathbb{R}^k$ with $f(y) \in U$, $g(y) \in V$ and $U,V$ disjoint. (just take small enough balls)
As $y \in f^{-1}(U) \cap g^{-1}(V)$, where the latter intersection is an open set, since $f,g$ are continuous, there has to be, due to $D$ being dense, a $\tilde{y} \in D \cap f^{-1}(U) \cap g^{-1}(V)$. This entails $f(\tilde{y}) = g(\tilde{y})$.
Thus $f(\tilde{y}) \in U\cap V$, which contradicts that $U$ and $V$ were disjoint.
Konstantin
- 628