Let $f, g : X \to Y$ be continuous functions. Assume that $Y$ is Hausdorff and that there exists a dense subset $D$ of $X$ such that $f(x) = g(x)$ for all $x \in D$. Prove that $f(x) = g(x)$ for all $x \in X$. Here is what I have so far, Proof: Let $f : X \to Y$ and $g : X \to Y$ be continuous ...

Akhil showed that the Cardinality of set of real continuous functions is the same as the continuum, using as a step the observation that continuous functions that agree at rational points must agree everywhere, since the rationals are dense in the reals. This isn't an obvious step, so why is it ...

$$ \textbf{PROBLEM} $$ Suppose $f$ and $g$ are two continuous functions such that $f: X \to Y $ and $g : X \to Y $. $Y$ is a a Hausdorff space. Suppose $f(x) = g(x) $ for all $x \in A \subseteq X $ where $A$ is dense in $X$, then $f(x) = g(x) $ for all $x \in X $. $$ \textbf{ATTEMPT...

Let $X$ be a topological space and let $Y$ be a Hausdorff space. Let $D$ be dense in $X$. Prove that continuous functions $f, g : X \to Y$ which are equal in $D$ are equal in all $X$. I'm a little stuck with this elementary proof. All help appreciated :)

The following problem is from Munkres's Topology (exercise 13 in section 18 "Continuous Functions"; page 112, 2nd edition). Let $A \subset X$; let $f:A \to Y$ be continuous; let $Y$ be a Hausdorff. Show that if $f$ may be extended to a continuous function $g:cl(A)\to Y$, then $g$ is uniquely...

There is a thread that is used for reopen requests Requests for Reopen & Undeletion Votes, etc. Some users do not use this thread for their questions on closed or deleted questions. A main reason could be ignorance about the existing thread; the question is not even tagged faq nor is there any...