Question 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.
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Check continuity at a point via limit equals value Classify common discontinuities on graphs Apply IVT to guarantee an intermediate output without constructing it Separate continuity from differentiability (corners can be continuous)
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Greenhouse temp 6 a.m.: 58°F Noon: 79°F Must it hit 70°F? Hole? Jump? Corner? Smooth?
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