Question

Note that f(x) is defined for every real x, but it has no roots. That is, there is no x∗ such that f(x∗) = 0. Nonetheless, we can find an interval [a, b] such that f(a) < 0 < f(b): just choose a = −1, b = 1. Why can’t we use the intermediate value theorem to conclude that f has a zero in the interval [−1, 1]?

Answer 1

Answer: Hello there!

Things that we know here:

f(x) is defined for every real x

f(a) < 0 < f(b), where we assume a = -1 and b = 1

and the problem asks: "Why can’t we use the intermediate value theorem to conclude that f has a zero in the interval [−1, 1]?

The theorem says:

if f is continuous in the interval [a, b], and f(a) < u < f(b), there exist a number c in the interval [a, b], such f(c) = u

Notice that the function needs to be continuous in the interval, and in this case, we don't know if f(x) is continuous or not, so we can't apply this theorem.