Understanding Intermediate value theorem. I will first state the theorem in my words and then my problem. > If function $f$ is continuous on $[a,b]$ and $f(a) > 0 > f(b)$ or $f(a) < 0 < f(b)$ then the function has **at least** a zero in $[a,b]$. * * * I have two problems with this theorem. 1. My book does not say that " **at least** " part, it just says "a zero", is not the book wrong ? example : $f(x) = x^3 - 5x^2 + 6x$ on $[-100,100]$. 2. Let $f$ be such that $f(x) = x^2$. Now $f$ is certainly continuous on $[-1,1]$ but $f(-1) = f(1) = 1$. Thus it does not satisfy any of the last two conditions still it has a zero at $x = 0$. What have I missed ?

You are misunderstanding the statement of the theorem. The first issue is just a grammar convention. Here, the phrase "there is a zero in $[a,b]$" implicitly means "at least one zero."

Second, this statement is _not_ biconditional. That is, the statement is of the form

$$\text{hypothesis} \implies \text{ there is a zero}$$

which does NOT mean

$$\text{there is a zero} \implies \text{hypothesis}.$$

If you could use the theorem in the reverse direction, we write

$$\text{hypothesis} \iff \text{ there is a zero}.$$

The forward arrow is an "if then" statement, and the double-headed arrow is an "if and only if" statement.