We extend classical results by Debreu and Dierker about equilibrium prices of a regular economy with continuously differentiable demand functions/excess demand function to a regular exchange economy with these functions being locally Lipschitz. Our concept of a regular economy is based on Clarke’s concept of regular value and we show that such a regular economy has a finite, odd number of equilibrium prices, the set of economies with infinite number of equilibrium prices has Lebesgue measure zero and there exist locally Lipschitz selections of equilibrium prices around a regular economy.