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true or false- continuous functions
I'm having some hard time deciding if those sentences are true or false:
$1$. If $f$ is continuous on $\mathbb{R}$ then if $\left|f(x)-x\right|<1$ for every $x$ on $\mathbb{R}$ then $f$ is getting ...
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Is $\frac{1}{\frac{1}{x}}$ defined at $x=0$?
In the context of projectively extended real line $\widehat{\mathbb{R}}$, if $f(x)=\frac{1}{\frac{1}{x}}$, then
$$f(0)=\frac{1}{\frac{1}{0}}=\frac{1}{\infty}=0.$$
But in the context of $\mathbb{R}$, ...
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