# Prove strict monotonicity of utility function

I have the following utility function: $$u(x_1, x_2, x_3) = med(x_1, x_2, x_3)$$ Given that $$UMG_{i}$$ ≥ 0, the utility function represents a strictly monotonic preference. Does this assertion make sense?"

I guess by $$med$$ you mean the median value. For example: $$med(2,7,19) = 7$$.
The sufficient condition you meant is $$UMG_i > 0$$.
Also, here we have $$UMG_i \geq 0$$ wherever it is defined, but there are points where $$UMG_i$$ isn’t defined as the function is not differentiable where at least two of the $$x_i$$’s are equal.