$f(x)=\begin{cases} 0 & \text{ if } x\in (-\infty,0) \\ 1 & \text{ if } x\in [0,+\infty] \end{cases}$, continuous?

Question

Question:

is $f: (\mathbb{R},τ) \rightarrow (Y,σ)$ with $f(x)=\begin{cases} 0 & \text{ if } x\in (-\infty,0) \\ 1 & \text{ if } x\in [0,+\infty] \end{cases}$, continuous?

Where $τ$ is the usual topology of $\mathbb{R}$, $Y=\{0,1,2\}$ and $σ=\{\emptyset,Y,\{0\}\}$

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my answer would that its continuous since $f^{-1}(\emptyset) \in τ$, $f^{-1}(Y)\in τ$ and $f^{-1}(\{0\})= (-\infty,0)\in τ$

Is my answer correct? If not, how can I properly answer and prove it?

Answer 1

Yes, your answer is correct. Recall that a map $f:(X,\tau)\to (Y,\sigma)$ is continuous iff for any $V\in \sigma$, $f^{-1}(V)\in \tau$. And that is exactly what you proved (as some people said in the comments, you need to replace "$\subset \tau$" by "$\in\tau$"), but nothing more is needed.