What is what does it mean for a function to be differentiable?

A function is said to be differentiable at a point if its derivative exists at that point. In simpler terms, this means that the function has a well-defined tangent line at that point.

Here's a breakdown of what that entails:

• Existence of the Derivative: The derivative of a function f(x) at a point x = a, denoted as f'(a), is defined as the limit:

  f'(a) = lim (h→0) [f(a + h) - f(a)] / h

  For a function to be differentiable at x = a, this limit must exist and be finite.

• Smoothness: Differentiability implies that the function is "smooth" at the point. This means there are no sharp corners, cusps, or vertical tangents at that point.

• Continuity: A function must be continuous at a point to be differentiable at that point. However, the converse is not true; a function can be continuous but not differentiable (e.g., the absolute value function at x=0).

• Left and Right-Hand Derivatives: The left-hand derivative and the right-hand derivative must exist and be equal at the point for the function to be differentiable. The left-hand derivative is the limit as h approaches 0 from the left, and the right-hand derivative is the limit as h approaches 0 from the right.

• Geometric Interpretation: Geometrically, differentiability at a point means that you can draw a unique tangent line to the graph of the function at that point. The slope of this tangent line is the value of the derivative at that point.

• Implications: If a function is differentiable on an interval, it implies certain properties like the applicability of theorems such as the Mean Value Theorem.

In summary, differentiability is a stronger condition than continuity. It requires the function to be continuous and "smooth" enough to have a well-defined tangent line at a particular point.