PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 42, Number 1, January 1974 CONTINUITY OF CERTAIN CONNECTED FUNCTIONS AND MULTIFUNCTIONS MELVIN R. HAGAN1 Abstract. In this paper it is proved that if Jfis a 1st countable, locally connected, 7Vspace and Y is a <7-coherent, sequentially compact TVspace, then any nonmingled connectedness preserving multifunction from X onto Y with closed point values and connected inverse point values is upper semicontinuous. It follows that any monotone, connected, single-valued function from Xonto Y is continuous. Let A" be as above and let y be a sequentially com- pact Tj-space with the property that if a descending sequence of connected sets has a nondegenerate intersection, then this inter- section must contain at least three points. If/is a monotone con- nected single-valued function from JTonto Y, then/is continuous. An example of a noncontinuous monotone connected function from a locally connected metric continuum onto an hereditarily locally connected metric continuum is given. In [1] and [2] conditions are given under which an open monotone connected function is continuous. This paper is concerned with conditions under which a monotone connected function is continuous. As Example 2 below shows, a monotone connected function from an hereditarily locally connected metric continuum onto a nonlocally connected metric continuum is not necessarily continuous, and Example 3 shows that a monotone connected function from a locally connected metric continuum onto an hereditarily locally connected metric continuum is not necessarily con- tinuous. It is an open question as to whether or not such a function is con- tinuous if both the domain and range are hereditarily locally connected metric continua. Some definitions will now be recalled. A multifunction F: X->- Y is upper semicontinuous at a point p e X if for any open set F<= Y, with F(p)<= V, there is an open set £/<= X, with/» e U, such that F((/)<= V, and Fis non- mingled provided for any p,qe X, either F(p)=F(q) or F(p)r\F{q)= 0. Received by the editors August 25, 1972. AMS (MOS) subjectclassifications (1970). Primary 54C10,54C60; Secondary54F20, 54F50. Key words and phrases. Upper semicontinuous multifunction, connectedness pre- serving function, monotone function, locally connected, a-coherent, hereditarily locally connected continuum. 1 Research supported by North Texas State University Faculty Research Grant No. 34590. © American Mathematical Society 1974 295 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use