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**dwsmith** Notation:

$\displaystyle (\mathbb{R},\tau_s)$ Standard topology on R

$\displaystyle (\mathbb{R}.\tau_l)$ Lower limit topology on R

Prove $\displaystyle f:\mathbb{R}\to\mathbb{R}$ is right continuous on $\displaystyle \mathbb{R}$ iff. f is continuous on $\displaystyle \mathbb{R}_l$ when considered as a function from $\displaystyle \mathbb{R}_l\to\mathbb{R}$.

$\displaystyle (\Rightarrow)$

Since f is right continuous on R, we know $\displaystyle \lim_{x\to x^+_0}f(x)=f(x_0)$

$\displaystyle f^{-1}((a_i,b_i))=\cup_{i\in I}[a_i,b_i)$

Where $\displaystyle (a_i,b_i)$ is open in X. I am not sure about the saying the invers equals the union of half closed and open sets. I am sort of lost.