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# The Problem

Let $$f:(0,\infty)\to\mathbb{R}$$ be a continuous function such that for all $$x\in(0,\infty)$$, $$f(2x)=f(x)$$Show that the function $$g$$ defined by the equation $$g(x)=\int_{x}^{2x} f(t)\frac{dt}{t}~~\text{for}~x>0$$is a constant function.

# Key Ideas

• One way to check if a differentiable function is constant is to check whether the derivative of the function is 0 everywhere.