Proof about a real-valued continuous function?? 1. The problem statement, all variables and given/known data Five line segments meet at a point. Show that any continuous real-valued function defined on this set must take the same value three times. 3. The attempt at a solution Take the values f(0,0,0) f(0,0,1) f(1,0,0) f(0,1,0) f(-1,0,0) f(-1,0,0). If three of the values are the same, the proof is done. Assume all of the values are different. At least three of f(0,0,1) f(1,0,0) f(0,1,0) f(-1,0,0) f(0,-1,0) are greater than or less than f(0,0,0). Without loss of generality, assume f(-1,0,0) < f(0,0,0) < f(0,1,0) < f(0,0,1). Then, by the Intermediate Value Theorem, there is a point p1 between f(-1,0,0) and f(0,0,1). Then p would be on the line segment from (-1,0,0) to (0,0,1) and also on two other line segments...but which?? Hence, the real-valued function takes the same value three times.