The Problem


Let \(f:\mathbb{R}\to\mathbb{R}\) be a continuous function such that for all \(x\in\mathbb{R}\) and for all \(t\geq 0\), $$f(x)=f(e^tx)$$Show that \(f\) is a constant function.

Key Ideas


  • Set \( \frac{x_2}{x_1} = t \) for all \( x_1, x_2 > 0 \). Do the same for x values less than 0.
  • Use continuity at x = 0

Discussion


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