Prove or disprove that $$ \{x: 2x^2\cos(1/x) \le 1\} \cup \{0\}\ \text{is complete}. $$
Define $$f(x)=\begin{cases} 2x^2\cos\left(\frac{1}{x}\right), & x\neq 0\\ 0, & x=0 \end{cases}$$ Clearly, $f$ is continuous and $T=f^{-1}\big((-\infty,1]\big)$ which is closed and hence $T$ is complete.
Am I right?