Let I be an open interval and f : I → ℝ is a function. How do you define "f is continuous on I" ?(adsbygoogle = window.adsbygoogle || []).push({});

would the following be sufficient? :

f is continuous on the open interval I=(a,b) if [itex]\stackrel{lim}{x\rightarrow}c[/itex] [itex]\frac{f(x)-f(c)}{x-c}[/itex] exists [itex]\forall[/itex] c[itex]\in[/itex] (a, b)

is this correct?

Also, what about the case of a closed interval I? In that case, can you just add to the above statement that:

[itex]\stackrel{lim}{x\rightarrow}a^{+}[/itex] f(x) = f(a)

and

[itex]\stackrel{lim}{x\rightarrow}b^{-}[/itex] f(x) = f(b)

???

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# Rigorous definition of continuity on an open vs closed interval

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