Continuity and Differentiability of Functions A function's continuity defines how its behavior behaves as its input changes. A function is continuous...
A function's continuity defines how its behavior behaves as its input changes. A function is continuous at a point if its limit is defined and equal to the function value at that point. This means the function approaches the function value as the input approaches that point.
For example, consider the function (f(x) = \frac{1}{x}) for (x \ne 0). This function is continuous at (x = 0) because its limit is defined and equal to 1, which is equal to the function value at (x = 0).
Differentiability measures how quickly a function's output changes with respect to its input. A function is differentiable at a point if its derivative exists and is finite. The derivative represents the instantaneous rate of change of the function at that point.
For example, consider the function (f(x) = x^2 + 1) for all (x \in \mathbb{R}). The derivative of this function is (f'(x) = 2x), which is a finite and continuous function for all (x).
| Feature | Continuity | Differentiability |
|---|---|---|
| Definition | Limit is defined and equal to the function value | Derivative exists and is finite |
| Intuitive Interpretation | Function approaches the function value as input approaches that point | Derivative represents the instantaneous rate of change of the function |
| Example | (f(x) = \frac{1}{x}) is continuous at (x = 0) | (f(x) = x^2 + 1) is differentiable at (x = 0) |