f:R ---> R is a continuous function and f(x) = f(2x) is true for all real numbers If f(1)=3 then find the value of
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Originally Posted by grgrsanjay f:R ---> R is a continuous function and f(x) = f(2x) is true for all real numbers If f(1)=3 then find the value of Necessary condition: Taking into account that is continuous in and satisfies for all then, if the solution does not depend on we necessarily have .
We can show that f is necessarily constant. Since for all x, .....(just substitute x/2 for x) ..... similarly ... Now let and apply the continuity of f.
Another way: the solutions of the functional equation are where is any periodic function with period . According to the hypothesis, necessarily .
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