In mathematics, a bump function (also called a test function) is a function on a Euclidean space which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump functions with domain forms a vector space, denoted or The dual space of this space endowed with a suitable topology is the space of distributions. Examples The function given by is an example of a bump function in one dimension. It is clear from the construction that this function has compact support, since a function of the real line has compact support if and only if it has bounded closed support. The proof of smoothness follows along the same lines as for the related function discussed in the Non-analytic smooth function article. This function can be interpreted as the Gaussian function scaled to fit into the unit disc: the substitution corresponds to sending to A simple example of a (square) bump function in variables is obtained by taking the product of copies of the above bump function in one variable, so Existence of bump functions It is possible to construct bump functions "to specifications". Stated formally, if is an arbitrary compact set in dimensions and is an open set containing there exists a bump function which is on and outside of Since can be taken to be a very small neighborhood of this amounts to being able to construct a function that is on and falls off rapidly to outside of while still being smooth. The construction proceeds as follows. One considers a compact neighborhood of contained in so The characteristic function of will be equal to on and outside of so in particular, it will be on and outside of This function is not smooth however. The key idea is to smooth a bit, by taking the convolution of with a mollifier. The latter is just a bump function with a very small support and whose integral is Such a mollifier can be obtained, for example, by taking the bump function from the previous section and performing appropriate scalings. An alternative construction that does not involve convolution is now detailed. Start with any smooth function that vanishes on the negative reals and is positive on the positive reals (that is, on and on where continuity from the left necessitates ); an example of such a function is for and otherwise.