Problems

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Found: 40

The function f(x) is defined and satisfies the relationship (x1)f((x=1)/(x1))f(x)=x for all x1. Find all such functions.

Is there a bounded function f:RR such that f(1)>0 and f(x) satisfies the inequality f2(x+y)f2(x)+2f(xy)+f2(y) for all x,yR?

On a function f(x) defined on the whole line of real numbers, it is known that for any a>1 the function f(x) + f(ax) is continuous on the whole line. Prove that f(x) is also continuous on the whole line.

A continuous function f(x) is such that for all real x the following inequality holds: f(x2)(f(x))21/4. Is it true that the function f(x) necessarily has an extreme point?