Let $T$ be the function defined by letting $T(x)$ be the decimal expansion of $x$ truncated to five decimal places, where $x$ is some real number. So, for example:
- $T(1/2)=0.5$
- $T(π)=3.14159$
- $T(1/3)=0.33333$
- $T(1)=1$
Let $ϵ=10^{−5}$. Show that for every $δ>0$ there is some $x$ such that $|x−\frac{1}{2}|<δ$ and $|T(x)−\frac{1}{2}|≥ϵ$.
I've been given the hint that the numbers $0.4999…99$, with $n$ number of $9$s, are very close to $\frac{1}{2}$ for large values of $n$.
I don't know where to start, I've tried finding $δ$ the normal way with $\epsilon$ but there isn't a clear function so I'm confused.