Originally Posted by

**slevvio** Hi everyone I was wondering if anybody could help me with this question; I have done the first 2 parts and a seperate 4th part but part 3 is causing me mathematical pain.

Suppose that $\displaystyle f: \mathbb{R} \rightarrow \mathbb{R} $

satisfies $\displaystyle f(t/2) = f(t)/2, t \in \mathbb{R} $.

1) Find f(0).

[easy to show that f(0) = 0 ]

2) Show that $\displaystyle f(2^{-n} t) = 2^{-n}f(t) \forall t \in \mathbb{R} \forall n \in \mathbb{N} $

[can be done easily by induction]

Now suppose that f is differentiable at 0.

3) Show that f(t) = t f'(0) for all $\displaystyle t \in \mathbb{R} $ [Hint: For fixed t, consider the sequence $\displaystyle \{ f(2^{-n}t) / (2^{-n} t) \} $

OK, we know that f is differentiable at 0, so

$\displaystyle \lim_{t \rightarrow 0} \frac{f(t)}{t} = f'(0) $

and for a fixed t, the above sequence simplifies to $\displaystyle \{ \frac{f(t)}{t} \} $ which is a constant sequence, but how does this help me? I guess I can say that

$\displaystyle \lim_{n \rightarrow \infty} \frac{f(t)}{t} = \frac {f(t)}{t} $

Any help would be appreciated :)