  We construct non-unique weak solutions $\theta\in C_t^0C_x^{0-}$ for forced
surface quasi-geostrophic (SQG) equation. This is achieved through a convex
integration scheme adapted to the sum-difference system of two distinct
solutions. Without external forcing, non-unique weak solutions $\theta$ in
space $C_t^0C_x^{\alpha}$ with $\alpha<-\frac15$ were constructed by
Buckmaster, Shkoller and Vicol, and Isett and Ma.
