I am using numerical methods to solve Schrodingers equation. I have identified an interval for E (energy) in which one solution tends to infinity, but on the other side the other solution tends to negative infinity. (see https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/MIT8_04S16_LecNotes12.pdf page 12, for example).
But I don't understand why there must be a solution in this interval? Why can't all values of E give solutions which rise to positive/negative infinity?