Let f (x) be continuous on [0,1] and f (0) = f (1). It is proved that there must be XO ∈ [0,1 / 2] such that f (XO) = f (XO + 1 / 2)

Let f (x) be continuous on [0,1] and f (0) = f (1). It is proved that there must be XO ∈ [0,1 / 2] such that f (XO) = f (XO + 1 / 2)

Consider the auxiliary function g (x) = f (x) - f (x + 1 / 2), 0

It is known that the function f (x) is continuous on [0,1], differentiable in (0,1), and f (1) = 0. It is proved that at least one point XO belongs to (0,1), such that f '(XO) = - KF (XO) / x0, K belongs to n

Consider the auxiliary function g (x) = (x ^ k) * f (x); use Rolle's theorem for G (x)

It is proved that if the function f (x) is continuous at the point XO and f (XO) ≠ 0, then there exists a neighborhood U of XO, if x ∈ u, f (x) ≠ 0

Let f (XO) = a ≠ 0, because the function f (x) is continuous at the point XO, so if limf (x) = a, e = | a | / 3 > 0, then there is a neighborhood U of XO;
||f(x)|-|a||