Continuity

A function is continuous at $a$ when ${\lim }_{x\to a}f(x)=f(a)$: the left-hand limit, right-hand limit, and function value must all agree. For piecewise functions, check the boundary point where the definition changes. Differentiability at $a$ requires continuity plus ${\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$ existing, meaning the left and right derivatives match. If a function is differentiable at every point, it is continuous at every point, but continuity alone does not guarantee differentiability.