Exercise · ZW-CONTINUOUS-MAP-TOPOLOGICAL-005 · ★★☆ Intermediate
In usual \(\mathbb{R}\), which of these functions \(f:\mathbb{R}\to\mathbb{R}\) is continuous?
Choose one option.
Show answer
- A. \(f(x)=x^2+1\)
Explanation
The other three have jumps. For instance, for \(\lfloor x\rfloor\) we get \(f^{-1}\big((\tfrac12,\tfrac32)\big)=[1,2)\), which is not open; for the indicator of \([0,\infty)\), \(f^{-1}\big((\tfrac12,\tfrac32)\big)=[0,\infty)\), also not open.
Review the concept: Continuous function between topological spaces
A function \(f:X\to Y\) between topological spaces \((X,\mathcal{T}_X)\) and \((Y,\mathcal{T}_Y)\) is continuous if the preimage of every open set \(V\) of \(Y\) is an open set of \(X\).
\[V\in\mathcal{T}_Y\ \Longrightarrow\ f^{-1}(V)\in\mathcal{T}_X\]
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