Semicontinuous function encyclopedia of mathematics. These functions gives us some new insights into how we can try to nd extreme values of functions even when there is no compactness. Finite refers to the number of players and the actions they have to choose. In particular, we prove that a function is lower semicontinuous if and only if its epigraph is a closed set. We define the notions of lower and upper semicontinuity for functions from a metric space to the extended real line. It is wellknown that all real function have a lower semicontinuous l. It is therefore closed, if and only if epi f is closed. Xr, for some topological space x, is lower semicontinuous if and only if it is continuous with respect to the scott topology on r. On minimax theorems for lower semicontinuous functions in. Lower and upper regularizations of frame semicontinuous real functions. For an interesting class dbsc differences of bounded semicontinuous see. Chapter 4 semicontinuities of multifunctions and functions.
This study was continued for vector valued functions in 2 and obviously, the. More upperlower semicontinuous functions in algebraic. The functional above is sequentially lower semicontinuous with respect to the uniform convergence in and the weak convergence in p, as a consequence of reshetnyaks lower semicontinuity theorem see, e. E is lower semicontinuous if and only if e is open. Lower semicontinuous functions with values in a continuous. We say that fis lower semi continuous at x 0 if for every 0 there exists 0 so that fx fx 0 1 whenever kx 0 xk 0 a number. The functional f is lower semicontinuous if and only if epif is closed. The value function of mayers problem arising in optimal control is investigated, and lower semicontinuous solutions of the associated hamiltonjacobibellman equation are defined in three equivalent ways. Semicontinuity of rank and nullity and some consequences. Then, the lower semicontinuity of f and are equivalent. The following theorem is the main result of this section. The definition can be easily extended to functions defined on subdomains of r and taking values in the extended real line. Recall that is said to be lower semicontinuous denoted by l.
Distance to spaces of semicontinuous and continuous functions. Second, we show that the psweak lower semicontinuity of a simple functional 1. A, respectively, by an arbitrary upper semicontinuous function and an arbitrary lower semicontinuous function. Pdf we investigate joint upper and lower semicontinuity of twovariable set valued functions.
R n r is lower semicontinuous if and only if its epigraph the set of points lying on or above its graph is closed. We summarize some basic properties of semicontinuous functions in the following proposition. On the upper and lower semicontinuity of the aumann integral nicholas c. One can easily verify that f is continuous if and only if it is both upper and lower semicontinuous. Lower and upper regularizations of frame semicontinuous. As a result, a lower semicontinuous convex function is the pointwise supremum of the affine functions that it majorizes. Yannelis university of illinois at urbanachampaign, champaign, il 61820, usa submitted september 1988, accepted july 1989 let t,r,p be a finite measure space, x. Some characterizations of error bound for nonlower. We see that the characteristic function of a set is lower semicontinuous if and only if the set is open. One might gain some insight into semicontinuity by showing that a set a. A novelty in mattilas work was the impossibility to use almgrens extrinsic bilipschitz embeddings of the space of qpoints into a euclidean space, because of the. The function jxjclearly has an absolute minimum over function fx jxjdoes.
The function f is said to be lower semicontinuous 1. In the cartesian graph of f1 shown in the left panel of figure 4. I believe that there is not a single coherent f giving the same stratification that the seshadri constant gives you could get it by a sequence of coherent f, or using noncoherent f. Minimax theorem for upper and lower semicontinuous payoffs. A b is continuous, if and only if it is upper semicontinuous as a correspondence.
Smoothness properties of the lower semicontinuous quasiconvex envelope marcus wagner 1. Let ff g 2j be a family of lower semicontinuous function in. Semicontinuous functions and convexity university of toronto. Existence of an equilibrium for lower semicontinuous. Let us introduce an important assumption of this paper.
We also give several equivalent characterizations of lower semicontinuity. We prove that a function is both lower and upper semicontinuous if and only if it is continuous. Pdf points of upper and lower semicontinuity for multivalued. A proof of the minimax theorem for the game played over a pair of compact metric spaces, in which the payoff is an upper or lower semicontinuous function. On minimax theorems for lower semicontinuous functions in hilbert spaces ewa m. Surely the vector space of first baire class is not exhausted by the cone of semicontinuous functions. Pdf 236 kb 2010 equivalent extensions to caristikirks fixed point theorem, ekelands variational principle, and takahashis minimization theorem. It is known and very easy to check that is upper semicontinuous and is lower semicontinuous. Then is an upper semicontinuous function, is a lower semicontinuous function, and we have that. Pdf lower semicontinuous functions semantic scholar.