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I'm having troubles solving the following exercise, proposed in Kosniowski's "A first course in Algebraic Topology", and any help would be appreciated!

Consider the topological space $X:=(\mathbb{R}, \mathcal{T})$, where $\mathcal{T}=\{\emptyset\}\cup \{\mathbb{R}\} \cup \{(-\infty, t):t \in \mathbb{R}\}$. Prove that a function $f: X \to X$ is continuous if and only if it is non-decreasing (that is, if $x > x'$, then $f(x)\ge f(x')$) and continuous on the right (that is, $\forall x \in \mathbb{R}$ and all $\epsilon > 0$ there exists $\delta > 0$ such that if $x' \in [x, x + \delta)$, then $|f(x) - f(x')| < \epsilon$).

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    Honestly, I really don't know what to do. Also a little hint could be sufficient in this moment...2012-12-22
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    Ok. First, write down what it means for a function between topological spaces to be continuous. Then use the definition of this topology to write down what this means in this specific case. Then try to connect this to the conditions you're given.2012-12-22

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