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The original question before it was marked as duplicated question:

I was wondering if one construct a non-continuous function $f:X\to Y$ between two topological spaces $X$ and $Y$, such that $f$ sends every convergent net into convergent net... I want to see an example...

Thanks in advance.

Edition:

Can Someone construct a non-continuous function $f:X\to Y$ between two topological spaces $X$ and $Y$, such that $f$ sends every convergent net into convergent net... I want to see an example...

Thanks in advance.

My own Answer:

Consider a two point set $X:=\{x,y\}$ with a topology $\mathcal{T}=\{\emptyset,\{x\},\{x,y\}\}$.

Define $f:X\to X$ by $x\mapsto y$ and $y\mapsto x$.

Then, since $f^{-1}(\{x\})=\{y\}$ where $\{x\}\in\mathcal{T}$, $f$ is not continuous.

On the other hand, pick any convergent net $\{x_\lambda\}_{\lambda\in\Lambda}$ in $X$.

We claim that $f(x_\lambda)$ converges to $y$.

Note that there is only one neighborhood of $y$, $\{x,y\}$, since $\{y\}$ is not open. Then, it follows that $\forall\mu\in\Lambda$, $\mu\prec\lambda\Rightarrow f(x_\lambda)\in\{x,y\}=X$. where $\prec$ is the preorder on the given directed set $\Lambda$.

User
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Maybe if we do not consider the limits. It is well-known that $f$ is continuous iff for every net $(x_i)$ that converges to $x$ in $X$, $f(x_i)$ converges to $f(x)$ in $Y$. So there would only be a potential example if the net need not converge to $f(x)$.

Henno Brandsma
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