$R(X,Y)f = 0$ holds for any connections?

Question

We define $R_{XY}$ for any connections $\nabla$ as following $$R_{XY}:=\nabla_{X}\nabla_{Y} - \nabla_{Y}\nabla_{X} - \nabla_{\left[X,Y\right]}$$ Then can we claim that $R_{XY} f = 0?$ since the following computation $$R_{XY}f = \nabla_{X}\nabla_{Y}f - \nabla_{Y}\nabla_{X}f - \nabla_{\left[X,Y\right]} f = \nabla_{X}(Yf) - \nabla_{Y}(Xf) - \left[X,Y\right](f) = X(Yf) - Y(Xf) - \left[X,Y\right](f) = 0.$$ I am not very sure about this conclusion.