I'm trying to show the uniqueness of the following SDE
$dX_t=\mu\left(t,X_t\right)X_{t}\ dt+\sigma\left(t,X_t\right)X_{t}\ dB_t$
where $B$ is a Brownian motion and $\mu, \sigma : \left[0,T\right] \times \mathbb{R}\rightarrow\mathbb{R}$ are bounded Borel functions, i.e., $\underset{\left[0,T\right]\times\mathbb{R}}{\text{sup}}\left(|\mu\left(t,x\right)|+|\sigma\left(t,x\right)|\right) < \infty$
Hint says there exists a unique solution for the SDE
$dX_t=a\left(t,X_t\right)dt+b\left(t,X_t\right)dB_t,\;X_0=x$
if$\;\exists\; C>0\; s.t. |a(t,x)-b(t,y)|+|b(t,x)-b(t,y)|\leq C|x-y| \quad \forall t\in[0,T],\ \forall x,y\in\mathbb{R}$
However, for the case $\mu\left(t,x\right)=\mathbb{1}_{\left[0,T\right]\times\mathbb{Q}}\left(t,x\right)$, which is an indicator function, the hint is not helpful, I think.
Because there is no other statements to prove the uniqueness of the solution for SDE I've learned in the lecture, I have no choice but using the hint.
I wonder what part of my thinking is wrong and how to solve it using hint. Thank you for your help.