Hi, I'm confused about an explanation in Elementary Calculus, the infinitesimal approach regarding Intermediate 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?