Continuation of a function

if I have a function $f(t)=e^{it}g(t)$, where $t=x\cdot \xi$ for $x,\xi \in R^n$. What are the conditions for defining $exp(it)= f(t)/g(t)$. The function $g$ is zero on a set with zero measure.

The exact expression is$\frac{1}{2\pi}\int_{\mathbb{R}}u(\xi\cdot x+s)\psi(\epsilon s)e^{-ias}ds = e^{ia\xi\cdot x}\frac{1}{\epsilon} \int_{-\epsilon}^{\epsilon} \hat u(a+t)\hat\psi\left(\frac{t}{\epsilon}\right)e^{i(\xi\cdot x)t}dt,$

where and $u\in L^\infty$ and $a\in R$ is a Lebesgue point for its Fourier transform $\hat{u}$ and $\psi$ is a Schwartz function whose Fourier transform $\hat\phi$ is supported in $[-1,1]$ satisfying $0\leq \hat\phi \leq 1$ and $\hat\phi = 1$ on $[-\frac{1}{2}, \frac{1}{2}]$.

Could someone help me determine the conditions under which $exp(it)= f(t)/g(t)$ holds?

Thank you in advance!

Best,Aristo