differentiability

Differentiability refers to whether a function has a well-defined, smooth slope at a specific point, meaning it can be closely approximated by a straight line near that point. If a function is differentiable at that point, it indicates no sharp corners, jumps, or breaks there. Think of it as the function’s ability to be "tangent" smoothly at that point, allowing us to measure its instantaneous rate of change accurately. Overall, differentiability ensures the function behaves in a predictable, smooth manner locally, facilitating analysis of its behavior and how it evolves around that point.