Continuity on [a,b][a, b][a,b] requires continuity on (a,b)(a, b)(a,b) plus:
Differentiability at the endpoints
One-sided continuity at the endpoints
f(a)=f(b)f(a) = f(b)f(a)=f(b)
f(a)=0f(a) = 0f(a)=0