Function
Two pieces glued at the join (dashed vertical line). The blue piece is the left formula, the red piece the right formula. The join is the only place anything can go wrong: each piece on its own is a polynomial and is perfectly smooth.
Continuity = the two pieces meet: left value equals right value at the join. One equation.
Differentiability = they meet and arrive at the same slope: left derivative equals right derivative at the join. A second equation.
Differentiable implies continuous. Continuous does not imply differentiable (a corner is continuous with mismatched slopes).
Exam recipe: "continuous at c" gives one equation from the values; "differentiable at c" gives that one plus a second from the derivatives. Two unknowns need both.