I have a Question about the following exercise:
Let $X,Y$ be topological spaces and $f:X \rightarrow Y$ in $x_0$ not continuous. Proof that there exists a net $(x_i)_{i \in I}$ in $X$ such that for $x_i \rightarrow x_0$, $(f(x_i))_{i \in I} \nrightarrow f(x_0)$ in $Y$.
I am not sure if this exercise is just extremely easy or if I misunderstood something. My Approach: Lets $f$ be like above and assume that for every net $(x_i)_{i \in I}$ in $X$ it follows that
(1) $x_i \rightarrow x$, $(f(x_i))_{i \in I} \rightarrow f(x)$ in Y.
We know that the condition (1) is equivalent to $f$ being continuous in $x$. So we have a contradiction.
This exercise really seemed too straight forward, and I would be thankful for feedback.