Consider two functions $x_1(t)$ and $x_2(t)$ as follows:
$$ x_1(t)=\left\{\begin{array}{ll} 0 & t<0 \\ 1 & t \geq 0 \end{array}\right. $$
$$ x_2(t)=\left\{\begin{array}{ll} 0 & t\leq 0 \\ 1 & t > 0 \end{array}\right. $$
Do they have same Fourier transforms?
As I know, the first one's Fourier transform is $F_1(\omega) = \pi \delta(\omega)+\frac{1}{j \omega}$. I know we can get this by calculating Fourier transform of $ \left\{\begin{array}{ll} e^{-\alpha t} & t \geq 0 \\ 0 & t<0 \end{array}\right. $ and finding its limit as $\alpha \to 0$.
I thought the Fourier transform of the second one should be the same. But I attended a Signals & Systems course and the lecturer said that there are some subtle difference between the two (But he didn't elaborated on the difference).
So my question is do their Fourier transform really differ? And if they don't have any difference, does this mean Fourier transform is not injective?