Fix an open subset of and denote the usual Euclidean norm by (so is endowed with the usual Euclidean metric). The following construction defines a smooth function that is positive on and vanishes outside of So in particular, if is relatively compact then this function will be a bump function. If then let while if then let ; so assume is neither of these. Let be an open cover of by open balls where the open ball has radius and center Then the map defined by is a smooth function that is positive on and vanishes off of For every let where this supremum is not equal to (so is a non-negative real number) because the partial derivatives all vanish (equal ) at any outside of while on the compact set the values of each of the (finitely many) partial derivatives are (uniformly) bounded above by some non-negative real number. The series converges uniformly on to a smooth function that is positive on and vanishes off of Moreover, for any non-negative integers where this series also converges uniformly on (because whenever then the th term's absolute value is ). As a corollary, given two disjoint closed subsets of and smooth non-negative functions such that for any if and only if and similarly, if and only if then the function is smooth and for any if and only if if and only if and if and only if In particular, if and only if so if in addition is relatively compact in (where implies ) then will be a smooth bump function with support in Properties and uses While bump functions are smooth, they cannot be analytic unless they vanish identically . This is a simple consequence of the identity theorem. Bump functions are often used as mollifiers, as smooth cutoff functions, and to form smooth partitions of unity. They are the most common class of test functions used in analysis. The space of bump functions is closed under many operations. For instance, the sum, product, or convolution of two bump functions is again a bump function, and any differential operator with smooth coefficients, when applied to a bump function, will produce another bump function. If the boundaries of the Bump function domain is to fulfill the requirement of "smoothness" it have to preserve the continuity of all its derivatives, which leads to the following requirement at the boundaries of its