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This is a solution of I.S.I 2018 Problem 3 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

## 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