Is there a (preferably elementary) proof that the graph of the (discontinuous) function $y$ defined on $[0,1)$ by
$$ y(x) =\begin{cases}
\sin\left(\dfrac{1}{x}\right) & \mbox{if $0\lt x \lt 1$,}\\\
0 & \mbox{if $x=0$,}\end{cases}$$
is not path connected?
General Topology – Topologist’s Sine Curve is Not Path-Connected
connectednessgeneral-topologypath-connected
Best Answer
If $S=\{(0,0)\}\cup\{(x,\sin(1/x)):0<x<1\}$ and $f=(f_1,f_2):[0,1]\to S$ is a path with $f(0)=(0,0)$, then $f(t)=(0,0)$ for all $t$.
To see this by contradiction, suppose that $f(t)$ is not always $(0,0)$. Removing an initial part of the interval and then rescaling if necessary, assume that $0=\sup\{t:f([0,t])=\{(0,0)\}\}$. By continuity of $f_2$, there is a $\delta>0$ such that $|f_2(t)|<1$ for all $t<\delta$. Take $t_0$ with $0<t_0<\delta$ and $f_1(t_0)>0$. By continuity of $f_1$ and the intermediate value theorem, $[0,f_1(t_0)]$ is in the image of $f_1$ restricted to $[0,t_0]$. Since $f_2(t)=\sin(1/f_1(t))$ for all $t$ with $f_1(t)\neq0$ and $\sin(1/x)$ maps $]0,\varepsilon[$ onto $[-1,1]$ for all $\varepsilon>0$, it follows that $[-1,1]$ is in the image of $f_2$ restricted to $[0,t_0]$. This contradicts $t_0<\delta$.