Let f be a continuous function on the interval I=[a,b] such that for every x in [a,b] f(x)≠0.
Show that the function f(x) doesn't change its sign.( like increasing or decreasing)
The Attempt at a Solution
Well for this to be true, we need to have f(a)>0 and f(b)>0 and f(x) is increasing so then it won't change the monotony. If we have f(a)<0 and f(b)<0, then f(x) is decreasing, hence we will not find any x in the interval I such that f(x)=0. Therefore for every x in the interval I f(x)≠0. Am I correct? Do I need to explain a bit more or what?