Continuity and differentiability are two closely related ideas in calculus, but they are not the same. A function is continuous at a point if its graph has no break, jump, or hole there. A function is differentiable at a point if it has a well-defined tangent slope there.
This distinction matters because many physical models require not only an unbroken graph, but also a meaningful rate of change.
Key Facts
- Continuity at x = a means lim(x -> a) f(x) = f(a).
- Differentiability at x = a means f'(a) = lim(h -> 0) [f(a + h) - f(a)]/h exists.
- If f is differentiable at x = a, then f is continuous at x = a.
- Continuity at x = a does not guarantee differentiability at x = a.
- A corner occurs when the left-hand and right-hand derivatives are finite but unequal.
- A cusp or vertical tangent can make the derivative fail to exist even when the graph is continuous.
Vocabulary
- Continuous function
- A function is continuous at a point if its value matches the limit of the function as x approaches that point.
- Differentiable function
- A function is differentiable at a point if it has a single finite derivative there.
- Derivative
- The derivative gives the instantaneous rate of change of a function with respect to its input.
- Corner
- A corner is a point on a continuous graph where the left and right slopes do not match.
- Cusp
- A cusp is a sharp point on a graph where the slope becomes undefined or changes too abruptly for a derivative to exist.
Common Mistakes to Avoid
- Assuming every continuous graph is differentiable, which is wrong because corners, cusps, and vertical tangents can be continuous but have no single tangent slope.
- Checking only that f(a) is defined, which is wrong because continuity also requires the limit as x approaches a to exist and equal f(a).
- Using the same rule on both sides of a piecewise point without checking one-sided derivatives, which is wrong because differentiability requires the left-hand and right-hand slopes to match.
- Calling a sharp corner a discontinuity, which is wrong because a graph can stay connected while still failing to have a derivative at that point.
Practice Questions
- 1 For f(x) = |x|, determine whether f is continuous at x = 0 and whether f is differentiable at x = 0. Find the left-hand and right-hand slopes.
- 2 Let f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x >= 1. Is f continuous at x = 1? Is f differentiable at x = 1?
- 3 A graph is unbroken at x = 3 but has a sharp corner there. Explain whether it is continuous, differentiable, both, or neither at x = 3.