We prove that a function is both lower and upper semicontinuous if and only if it is continuous. lower semicontinuous functions on R and prove some basic results like lattice prop-erties and some equivalent definitions. For. (c) Show that f is continuous at x 2D if and only if f is lower and upper semicontinuous at x. 5. The subdifferential ∂fof any proper convex lower semicontinuous function f: X→ ]−∞,+∞] is maximal monotone. replacing ´F and ´A, respectively, by an arbitrary upper semicontinuous function and an arbitrary lower semicontinuous function: A topological space X is normal if and only if, for any up-per semicontinuous f: X ! However, in the case of m≥ 2, condition (1.4) is not equivalent to the convexity of f; see Remark 3.3 below. The epigraph of a convex function is convex. Clearly, f is upper semicontinuous if and only if −f is lower semicontinuous, so it This result was announced by the author in a previous paper, but the argument given there was incomplete; the result is proved If (X;˝) is a topological space, then f : X ![1 ;1] is said to be lower semicontinuous if t2R implies that f1(t;1] 2˝. We say that f is \fnite if 1 < >: X t<0 A 0 \u0014t<1 ; t\u00151 We see that the characteristic function of a set is lower semicontinuous if and only if the set is open. a lower semicontinuous and convex function in Banach space implies continuity). If for some µ>0 and each x ∈ X with γ