Connected, but it is not continuous at some point(s) of I I have a mathematics problem, but I have no idea. please help me...
The problem is "Give an example of a function $f(x)$ defined on an interval I whose graph is connected, but is is not continuous at some point(s) of $I$"
In my idea, a solution is topologist's sin curve.
\begin{gather*}
f\colon[0, 1] \to [0, 1] \\\\
f(x) = \begin{cases}
\sin(1/x) &(x\neq 0)\\
0 &(x=0).
\end{cases}
\end{gather*}
Is this function connected on $\mathbb{R}^2$ space but not continuous at $x=0$ ?
Please help me...

This is a good example. The graph of the topologist's sine curve, which includes the point $(0,0)$ as you have indicated, is indeed connected. However, it is not continuous. To see this, try and produce a sequence of points $x_n$ converging to $0$ for which $\sin(1/x_n) = 1$.
