Hello. I have the following query. Consider a continuous function f(x). We want to know whether this function ever gets zero or not. So we make the assumption that it does. If (e.g) we have [itex]f(x)=x^2 + 4x + 2 - cosx[/itex] we'll assume that [itex]f(x)=0 (=) x^2 + 4x + 2 -cosx =0 [/itex] now we differentiate with respect to x [itex]2x + 4 +sinx =0[/itex] and again [itex]2 + cosx = 0 (=) cosx = -2 [/itex] which is impossible. Right now we have proven that f ''(x) can't be zero. Is there any theorem that, given some prerequisites for f(x), can show that f(x) can't be zero too, using the above demonstrated way? Thanks in advance, I hope you understand the core of my question. (I am not looking for the Bolzano Theorem, btw) Edit1: Another way of asking this is : Is there any theorem that proves, given prerequisites, that if [itex]f^n (x)=0 then f(x)=0[/itex], I won't specify how many (n) times differentiated.