In the semi linear uniform space

Question

In the semi linear uniform space, If $f$ is a function from $(X ,Γ_X)$ to ($Y,Γ_Y)$ where $f(x_n)$ converges to $f(x)$ whenever $x_n$ converges to $x$,show that $f$ is continuous at $x$.

Answer 1

Let $f$ be not continuous at $x$. There is some net $(d_t)_{t\in T}$ with $d_t \to x$ and $f(d_t)\not\to f(x)$. So we can assume there some net $(U_t)_{t\in T}$ of neighborhoods with $$(\forall t\in T)(f(t)\notin U_t)$$

As $\{U_t\mid t\in T\}$ is a chain, there must be a strictly decreasing sequence ${U_{t_n}}$ such that $$(\forall n)(f(t_n)\notin U_{t_n})$$ But then by assumption, $d_{t_n}\not\to x$. A contradiction.