A function is differentiable at x if it looks like a straight line near x.
Its derivative at x is the slope of that line.
It is continuous if it has no gaps.
These notions are defined formally with examples of their failure.
2.1 Informal Definitions of Limit and Continuity
2.1.3 Limits and Continuity
2.2 Formal Definitions
2.3 Examples of Discontinuities
2.4 Slope and Derivative
2.5 Differentiability and Continuity