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I´m thinking about why the distance for the set $\{f : [a,b] \to \mathbb{R} : \mbox{continious and bounded}\}$ is define as $$d_{\infty}(f,g) := sup_{x \in [a,b]}|f(x)-g(x)|$$ meanwhile for $\{f : [a,b] \to \mathbb{R} : \mbox{continious}\}$ is given as $$d(f,g) := \int_a^b|f(x)-g(x)| dx.$$

I know that you can assign any distance, of course. But my question is if they are well-defined.

What happens, for $0$ the zero constant map, with $d(0,\frac{1}{x}) := \int_a^b|\frac{1}{x}| dx$?

Thank you very much.

LH8
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1 Answers1

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Since your function $1/x$ is continuous on $[a,b]$, $0\notin [a,b] $ and the integral in $$d(0,\frac{1}{x}) := \int_a^b|\frac{1}{x}| dx$$ is well defined and there is no problem with this definition.

Mohammad Riazi-Kermani
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