Question

Помогите! Легкий вопрос! О функции f(x), заданной на всей вещественной прямой,известно, что при любом a > 1 функция f(x) + f(ax) непрерывна на всей прямой.Докажите, что f(x) также непрерывна на всей прямой.

Answer 1

Ответ:

f+e =8

8-f=e

e +x=f вроде так

Answer 2

Proof of Continuity of f(x) for f(ax)

To prove that the function f(x) is continuous on the entire real line, given that f(ax) is continuous for any a > 1, we can use the concept of a change of variable.

Let's assume that f(ax) is continuous for any a > 1. We want to show that f(x) is also continuous on the entire real line.

To do this, we can make a change of variable by substituting x = ax. This allows us to express f(x) in terms of f(ax).

Let's define a new function g(x) = f(ax). Now, we can rewrite f(x) as f(x) = g(x/a).

Since f(ax) is continuous for any a > 1, g(x) = f(ax) is also continuous for any x. Therefore, g(x/a) = f(x) is continuous for any x.

Hence, we have shown that f(x) is continuous on the entire real line, given that f(ax) is continuous for any a > 1.

Conclusion

In conclusion, we have proved that if the function f(ax) is continuous for any a > 1, then the function f(x) is also continuous on the entire real line. This result is obtained by making a change of variable and expressing f(x) in terms of f(ax).