Hi, I'm confused about an explanation in(adsbygoogle = window.adsbygoogle || []).push({}); Elementary Calculus, the infinitesimal approachregardingIntermediate Value Theorem.

The explanation:

Suppose g is a continuous function on an inverval I, and g(x) # 0 for all x in I

(i) If g(c) > 0 for at least one c in I, then g(x) > 0 for all X in I

(ii) if g(c) < 0 for at least one c in I, Then g(x) < 0 for all x in I

Proof: Let g(c) >0 for some c in I. If g(x1) < 0 for some other point x1 in I, Then by Intermediate value theorem, there is a point x2 between c and x1 such that g(x2) = 0, contrary to hypothesis. Therefore we conclude that g(x) > 0 for all x in I

The one thing I don't understand is this:

If g(x) # 0 for all x in I, then how can there be a point x2 such that g(x2)=0 between c and x1 (assuming c and x1 and x2 are within the inverval I) if g(x) is # 0 for all x??

"g(x) is # 0 for all x"tells me that the equation g(x) can never be zero for any x right?

And how can we conclude that g(x) > 0 for all x in I, if there exists a point x2 such that g(x)=0?

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# Intermediate Value Theorem

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