After noticing that function $f: \mathbb R\rightarrow \mathbb R $ $$ f(x) = \left\{\begin{array}{l} \sin\frac{1}{x} & \text{for }x\neq 0 \\ 0 &\text{for }x=0 \end{array}\right. $$ has a graph that is a connected set, despite the function not being continuous at $x=0$, I started wondering, doest there exist a function $f: X\rightarrow Y$ that is nowhere continuous, but still has a connected graph?
I would like to consider two cases
- $X$ and $Y$ being general topological spaces
- $X$ and $Y$ being Hausdorff spaces
But if you have answer for other, more specific cases, they may be interesting too.
from Hot Weekly Questions - Mathematics Stack Exchange
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