# If a function f is differentiable then its derivate is Borel measurable

I need help solving this interesting result:

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be differentiable. Prove that $f'$ Borel measurable.

I tried to start with, if $f$ is differentiable then $f'(x)$ exists for all $x∈ \mathbb{R}$.

• Hint: $f'$ is a limit of measurable functions.
– J.R.
Commented Nov 12, 2016 at 17:29

$f'$ is the pointwise limit of the sequence:

$$f_n(x) = \frac{f(x+1/n) - f(x)}{1/n}$$

• Thank you, but how do you prove that $f_n(x)$ is measurable? And with that you have proved that $f'$ is measurable but not Borel measurable... Commented Nov 16, 2016 at 9:50
• @JennaTaylor the composition of Borel functions is a Borel function, this justifies that $f_n$ is Borel. Now the pointwise limit of Borel functions is also Borel.
– user384138
Commented Nov 16, 2016 at 12:38
• Yes, the composition of Borel measurable functions is Borel measurable. But how do you prove that each $f_n$ is Borel measurable? That is what I got left... Commented Nov 16, 2016 at 14:10

You could also say that: f=differentiable => f=continuous. Since f=differentiable => $$\exists$$f '(x) = $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$ => f '(x) = lim$$_n\frac{f(x+1/n)-f(x)}{1/n}$$.

Then from Heine's deffinition (?)* $$\forall$$(x$$_n$$)$$_n\in\mathbb{R}$$ and $$\epsilon$$>0: x$$_n$$ $$\rightarrow$$ x$$_0$$ $$\Rightarrow$$ $$\sigma$$(f(x$$_n$$),f(x)) < $$\epsilon$$.

Suppose f$$_n$$(x) = $$\frac{f(x+1/n)-f(x)}{1/n}$$, f=continuous (so f(x+1/n)=continuous $$\forall$$n). Therefore so far we have: $$\forall$$x>0 and $$\forall\epsilon$$>0 $$\exists$$n$$_0\in\mathbb{N}$$ so that as n$$\geq$$n$$_0$$ then $$\sigma$$(f(x$$_n$$),f(x)) < $$\epsilon$$ $$\Rightarrow$$ (f$$_n$$)$$_n$$ $$\rightarrow$$ f p.w.

Then f=Baire-1 => f=Borel-meas. (this because f=lim$$_n$$f$$_n$$ $$\Rightarrow$$ f=limsup$$_n$$f$$_n$$, for (f$$_n$$)$$_n$$ = seqeuence of meassurable functions)

*I dont know how it is called in English. In Measure Theory or Real Analysis we call it in Greek "Arxi/Archi tis Metaforas" (= principle of transport).

PS: I'm an undergraduate student (mathematics), therefore there may be many mistakes (in english and in mathematical structure) I hope i helped.

• What you call "Archi tis Metaforas" is called "Sequential Criteria" in English texts. Commented Feb 17 at 16:39
• Thank you, I didn't know that! Commented Feb 21 at 14:04