I was told that the Takagi function, $T:[0,1]\rightarrow \mathbb{R}$, is continuous and has uncountable dense level sets in $[0,1]$. This has confused me for the following reason:
Suppose $L$ is a dense level set of the Takagi function. Doesn't this mean that for any point $s\in[0,1]$ I can find a convergent set of points $\{s_n\}$ in $L$ that tend to $s$? If so, doesn't continuity imply that $T(s_n)$ tends to $T(s)$. But this would imply $s\in L$ for all $s \in [0,1]$, so the whole interval is in the level set. This can't be right surely?
Please could someone explain my misunderstanding?