The Taylor's theorem is a popular and widely used tool for approximating a function by a polynomial. This is only possible when the function possesses continuous derivatives up to a corresponding order. To be able to get a more precise formula for the remainder, i.e. the difference between the function and its Taylor polynomial, the standard theorem requires the function to have in addition one more continuous derivative. In this paper we bring a simple generalization of such result, allowing the highest order derivative to have jump discontinuities.