derivatives with respect to the same dominating measure.) The above discussion of the likelihood for discrete random variables uses the counting measure, under which the probability density at any outcome equals the probability of that outcome. Likelihoods for mixed continuous–discrete distributions The above can be extended in a simple way to allow consideration of distributions which contain both discrete and continuous components. Suppose that the distribution consists of a number of discrete probability masses and a density , where the sum of all the 's added to the integral of is always one. Assuming that it is possible to distinguish an observation corresponding to one of the discrete probability masses from one which corresponds to the density component, the likelihood function for an observation from the continuous component can be dealt with in the manner shown above. For an observation from the discrete component, the likelihood function for an observation from the discrete component is simply where is the index of the discrete probability mass corresponding to observation , because maximizing the probability mass (or probability) at amounts to maximizing the likelihood of the specific observation. The fact that the likelihood function can be defined in a way that includes contributions that are not commensurate (the density and the probability mass) arises from the way in which the likelihood function is defined up to a constant of proportionality, where this "constant" can change with the observation , but not with the parameter . Regularity conditions In the context of parameter estimation, the likelihood function is usually assumed to obey certain conditions, known as regularity conditions. These conditions are in various proofs involving likelihood functions, and need to be verified in each particular application. For maximum likelihood estimation, the existence of a global maximum of the likelihood function is of the utmost importance. By the extreme value theorem, it suffices that the likelihood function is continuous on a compact parameter space for the maximum likelihood estimator to exist. While the continuity assumption is usually met, the compactness assumption about the parameter space is often not, as the bounds of the true parameter values are unknown. In that case, concavity of the likelihood function plays a key role. More specifically, if the likelihood function is twice continuously differentiable on the k-dimensional parameter space assumed to be an open connected subset of there exists a unique maximum if the matrix of second partials