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Suppose that $f:[0,1]\to\mathbb{R}$ is differentiable on (0,1) and continuous on [0,1]. Would like to assert that if f(0)=0, and $|f'(x)|\leq |f(x)|$ for each $x\in (0,1)$, then f is the zero function.

I have tried applying several different variations of the mean value theorem, but nothing useful has come of this. I also have that the derivative has a max and min (based on the compact domain and the inequality condition). What am I missing here to get that $f=0$?

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    $\begingroup$ If $0<x<1$, there is a $c_1\in(0,x)$ with $|f(x)|\le |f(c_1)|x$ by MVT. Apply MVT again to obtain $|f(x)|\le|f(c_2)|c_1 x$ with $0<c_2<c_1$. Keep going... $\endgroup$ – David Mitra Dec 5 '14 at 9:47
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Denote by $M$ the quantity $\max_{t\in [0,1]}|f(t)|$. Using the mean value theorem we get for $x\in (0,1)$, $$|f(x)|\leqslant x M.$$ We can prove by induction that $|f(x)|\leqslant x^nM$ for each $x\in (0,1)$. Indeed, it is true for $n=1$, and if it is true for $n$, then by the mean value theorem, $$f(x)=f'(c)x $$ for some $c\in (0,x)$, hence $|f(x)|\leqslant x|f'(c)|\leqslant x|f(c)|$. Using the induction hypothesis, we get $|f(x)|\leqslant xc^nM\leqslant x^{n+1}M$.

Since $x^n\to 0$ as $n$ goes to infinity, we get the wanted conclusion.

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  • $\begingroup$ interesting. does this mean that the hypothesis f(0)=0 is unnecessary? $\endgroup$ – reluctant mathematician Dec 7 '14 at 16:51
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    $\begingroup$ oh, wait... it gets used in the quotient $|\frac{f(x)-f(0)}{x-0}|\leq M$ $\endgroup$ – reluctant mathematician Dec 7 '14 at 16:58
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$$ |f(x)|=\Bigl|\int_0^xf'(t)\,dt\Bigr|\le\int_0^x|f'(t)|\,dt\le\int_0^x|f(t)|\,dt. $$ Let $g(x)=\int_0^x|f(t)|\,dt$. Then $$ g(x)\ge0,\quad g(0)=0,\quad g'(x)-g(x)\le0. $$ Multiplynig the last inequality by $e^{-x}$ we get $$ \bigl(e^{-x}g(x)\bigr)'\le0\implies e^{-x}g(x)\le g(0)=0. $$

Actualy, this is a particular case of Gronwall's lemma.

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  • $\begingroup$ Don't you need to assume $f'$ is integrable here? $\endgroup$ – David Mitra Dec 5 '14 at 11:59
  • $\begingroup$ $f'$ is defined everywhere and bounded, so it is Lebesgue integrable and the Fundamental theorem of Calculus applies. $\endgroup$ – Julián Aguirre Dec 5 '14 at 13:54
  • $\begingroup$ Ah, thanks.${}$ $\endgroup$ – David Mitra Dec 5 '14 at 14:16
  • $\begingroup$ This proof even works for the more general condition $|f'(x)| \le K |f(x)|$ for some positive constant $K$. $\endgroup$ – Martin R Mar 2 '17 at 10:40

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