The definition of Fourier Integral seems wrong in my textbook. The definition in book goes like this,
If $f(x)$ is continuous and $\int_{-l}^{l}\left | f(x) \right |dx$ converges to zero as $l$ approaches to infinity, then $f(x)$ can be represented by $$f(x) = \int_{0}^{\infty} \left [ A(\omega)\cos\, \omega x + B(\omega) \sin\, \omega x\right ]d\omega$$ where, $$A(\omega) = \frac{1}{\pi} \int_{-\infty}^{\infty} f(x) \cos (\omega x )\, dx$$ $$B(\omega) = \frac{1}{\pi} \int_{-\infty}^{\infty} f(x) \sin (\omega x) \, dx$$
I can't understand one thing how can $\int_{-l}^{l}\left | f(x) \right |dx$ converge to zero when $l$ approaches $\infty$. Is't it only possible if $f(x)=0$ otherwise any function say, $f(x)$ have some part above or below X-axis and $\left | f(x) \right |$ integrated over an range > 0.
I asked my professor about it but I didn't get what he said can somebody explain ?