If f(c) = infinity and c is in [a,b]

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SUMMARY

If f(c) = infinity for c in the interval [a,b], then f is not continuous at c. This conclusion is supported by the fact that a continuous function on a closed interval [a,b] must be bounded. Specifically, if a function is continuous within this interval, it must possess both a superior and inferior bound, meaning there exist real numbers N such that f(x) ≤ N and f(x) ≥ N for all x in [a,b]. Therefore, the presence of infinity indicates that f is not defined at c, disqualifying it as a function in this context.

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if f(c) = infinity and c is in [a,b]
is it equivalent to saying
f is not cont. at c ? because infinity is undefined?


Thanks
k.cv
 
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It's NOT continuous.
 
SO it would also imply that if a function is cont. in a finite interval [a,b] then its bounded?
 
Yes, it has superior bound. There exist a number N such that f(x)<= N for every x in [a,b]. Geometricly speaking this means there exist a parallel line to the horizontal axis. And of course a inferior bound, the same for a number N such that f(x) >= N for every x in [a,b].
 
Last edited:
If a function f is continuous on a closed and bounded interval, then it is bounded. You implied "closed" when you said [a,b] but I want to make sure that is clear.
 
Last edited by a moderator:
rsnd said:
if f(c) = infinity and c is in [a,b]
is it equivalent to saying
f is not cont. at c ? because infinity is undefined?


Thanks
k.cv


If you're claiming f is a function from [a,b] to R, then f is not defined at c, and actually f therefore isn't a function, never mind a continuous one.
 
nice...I just invented the mean value theorom!
 

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