Mod note: Edited the function definition below to reflect the OP's intent.
Suppose f:R->R is continuous. Let λ be a positive real number, and assume that for every x in R and a>0,f(ax)=aλ f(x). (a) If λ > 1 show that f is differentiable at 0. (b) If 0 < λ < 1 show that f is not differentiable at 0. (c) If λ=1, show that f is differentiable at 0 if and only if it is linear. (Hint: what is f(0)?)
The Attempt at a Solution
I am considering start the question with f(x)-f(0)/x but how can i find the limit when x approaches 0? f(x)=f(1*x)=1^λf(x) f(0)=f(0*x)=0^λf(x)=0, then is f(x)-f(0)/x = f(x)/x? Then how can i know it's differentiable at 0?
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