- #1

Joschua_S

- 11

- 0

I have a mathematical problem that I could not solve. Could you please give me some hints how to solve it?

Let [itex]f: [0,1] \rightarrow \mathbb{R}[/itex] be a continuous and on [itex](0,1)[/itex] a differentiable function with following properties:

a) [itex]f(0) = 0[/itex]

b) there exists a [itex]M>0[/itex] with [itex]|f'(x)| \leq M |f(x)| [/itex] for all [itex]x \in (0,1)[/itex]

Now the problem is: Show that [itex]f(x) = 0[/itex] is true for all [itex]x \in [0,1][/itex]

There is a hint given but it doesn't help me The hint is: Consider the set [itex]D = \{ x \in [0,1]: ~ f(t) =0 [/itex] for [itex]t \in [0,x] \}[/itex] and show that the the supremum of this set is [itex]1[/itex].

Thanks for help

Greetings