I'm trying to solve $f'(t)=0$ and $f'(t)=1$ using Fourier transform, but no luck:
a) $f'(t)=0$
$$ f'(t)=0 \Rightarrow jwF(w)=0 \Rightarrow \begin{cases}F(w)=0 ~ \text{if} ~ w \ne 0\\ F(0) = \text{undefined ?} \end{cases} $$
Addendum from comments of @Winther (if I understood correctly): In case we choice $F(0) \ne \infty$ then $f(t)=0$ that is one of the valid solutions; if we choice $F(0) = \infty$ then $f(t)=1$, another valid solution; no idea how to obtain all other possible solutions.
b) $f'(t)=1$
$$ f'(t)=1 \Rightarrow jwF(w)=\delta(w) \Rightarrow \begin{cases}F(w)=0 ~ \text{if} ~ w \ne 0\\ F(0) = \text{impossible! because left part is} =0 \text{ but right is }=\infty \end{cases} $$
After this disaster, I've tried to proof the results (calc the derivative using Fourier transform) that I expect as the correct solutions:
c) $f(t)=c$ (being $c$ in $\mathbb{R}$)
$$f(t)=c \Rightarrow\\ F(w)=c\delta(w) \Rightarrow\\ \mathscr{F}\{f'\}(w) = jwc\delta(w) = 0 \Rightarrow \\ f'(t) = 0 $$
success
d) $f(t)=t+c$
$$f(t)=t+c\Rightarrow\\ F(w)=\delta^{(1)}(w)+c\delta(w)\Rightarrow\\ \mathscr{F}\{f'\}(w) = jw\delta^{(1)}(w)+jwc\delta(w) = jw\delta^{(1)}(w) = ~ ? $$
( where $\delta^{(1)}(x)=\frac{d}{dx}\delta(x)$ )
blocked
In the case $1$, if $F(\omega)=0$ for all $\omega\ne 0$ then $F(\omega)$ can be $k\delta(\omega)$ for any real $k$ therefore$$f(t)={k\over 2\pi}$$Case $2$
First of all, we prove the following interesting property of $\delta(.)$ and $\delta'(.)$: $$\omega\delta'(\omega)=-\delta(\omega)$$notice that $\omega\delta'(\omega)=0\quad,\forall\omega\ne 0$ so we need to prove that $$\int_{0^-}^{0^+}\omega\delta'(\omega)d\omega=-1$$which is obvious because$$\int_{0^-}^{0^+}\omega\delta'(\omega)d\omega=\omega\delta(\omega)|_{0^-}^{0^+}-\int_{0^-}^{0^+}\delta(\omega)d\omega=-1$$which completes our proof. Now we have$$f'(t)=1$$therefore$$F(f')(\omega)=2\pi\delta(\omega)=-2\pi\omega\delta'(\omega)+2\pi j k\omega \delta(\omega)$$therefore$$F(f)(\omega)={F(f')(\omega)\over j\omega}=2\pi j\delta'(\omega)+2\pi k\delta (\omega)$$therefore$$f(t)=t+2\pi k\quad,\quad k\in\Bbb R$$