Fuzzy systems are universal approximators for a smooth function and its derivatives

One of the reasons why fuzzy methodology is successful is that fuzzy systems are universal approximators, i.e., that we can approximate an arbitrary continuous function within any given accuracy by a fuzzy system. In some practical applications (e.g., in control), it is desirable to approximate not only the original function, but also its derivatives (so that, e.g., a fuzzy control approximating a smooth control will also be smooth). In our paper, we show that for any given accuracy, we can approximate an arbitrary smooth function by a fuzzy systems so that not only the function is approximated within this accuracy, but its derivatives are approximated as well. In other words, we prove that fuzzy systems are universal approximators for smooth functions and their derivatives.