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Let $a \geq 0$, and the application $T_a$ from $C(\mathbb{R}^+, \mathbb{R})$ with values in $\mathbb{R}^+\cup \{ \infty \}$ defined by $\inf \{ t \geq a , f(t) \geq a \}$. I for all $t \geq 0$, $f(t) < a$, then $T(f) = \infty$

I do not think this function is continuous. Indeed let $\varepsilon$ > 0, $f$ defined by $f(t) = a$ and $g(t) = a - \varepsilon$. Then $T_a(f) = 0$ and $T_a(g) = + \infty$.

But, I do not know if this function admits continuity points. Your opinion ?

Thanks in advance.

Jose Avilez
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user1240705
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    What kind of topology is given to $C(\mathbb{R}^+, \mathbb{R})$? This choice will certainly affect what it means by "continuity point of $T_a$". – Sangchul Lee Nov 10 '23 at 22:47
  • Thanks. I think $C(\mathbb{R}^+,\mathbb{R})$ is equipped with the norm$.||.||_{\infty}$ – user1240705 Nov 11 '23 at 12:40
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    Please don't put clarifications and additional information into the comments. Instead, revise your post so that it incorporates all relevant information and reads well for someone who encounters it for the first time. – D.W. Nov 13 '23 at 06:05

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