Owl: A greenhouse thermometer logs every few seconds from 6 a.m. to noon. At dawn it's 58°F; by noon it's 79°F. Between those readings, could the continuous temperature curve somehow skip 70°F entirely?
Intuition says no — if the graph never lifts the pen, every in-between height has to appear at least once. That's the Intermediate Value Theorem whispering.
Continuity itself means: as you approach a point, the outputs approach the function's actual value there — no holes, no jumps, no wild breaks.
Careful though: a sharp corner on a graph can still be continuous. Continuity is not the same as differentiability. Where should we start — checking a point, or classifying whole graphs?