Hi,
If you look at your signal from minus inf. to inf. you see a frequency spectrum for it. it's the frequency response of your whole signal. the question is, can we see a sudden changing of local frequency in t=0 directly with my whole signal spectra and interpret it. your answer is no. but what is solution?. the main general approach for solving this problem is to look (ans process) our signal locally and with shifting in Time. => Time-Frequency Transforms.
one of solution is Short Time Fourier Transform (STFT). Ther are some other solutions like Wigner-Ville Trans., Wavelet Trans. (CWT & DWT).
Regards