Three problems concerning ideals of differentiable functions

In this paper, we study the validity of the following two statements in the internal logic of the toposes of Synthetic Differential Geometry: 1. (1) The integral of f is non-negative if f is non-negative; 2. (2) If f=0 in the set of non-negative reals, and f=0 in the set of non-negative reals, then f=0. We find statements (1) and (2) to be true in the toposes considered. We also prove that 3. (3) For n greater than two, the arrow tn from the line to itself is not a stable effective epic. This answers a question raised by Quê-Moerdijk-Reyes.