A map X -- IR from a space X is said to be locally constant if for each x E X there is some open set U with x E U and flU constant. Prove that if X is connected then every locally constant map f : X -- IR is constant. [s this still true if IR is replaced by an arbitrary topological space

Astudent is determining the influence of different types of food on the growth rate of spiders. she feeds one group of spiders two caterpillars twice a week. the other group received similarly-sized beetles twice a week. she records the amount of time required for individuals to reach sexual maturity. her hypothesis is that the spiders feeding on caterpillars will have a faster growth rate because caterpillars have a higher protein content compared to beetles. in this experiment, what is the dependent variable? spiders assigned to different feeding groups. time required to reach sexual maturity number of prey items fed twice a week. growth rates of each group.