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.
Understanding Calculus: Differentiability vs Continuity
A useful way to separate these ideas is to zoom in on the graph. Near an ordinary smooth point, a tiny piece of the curve becomes almost indistinguishable from a straight line. The slope of that local line is the derivative.
To estimate it, calculus compares the slopes of lines joining the point to nearby points. Those nearby slopes must settle toward one number as the gap shrinks. At a sharp corner, the slope from the left can settle at one value while the slope from the right settles at another.
The graph of the absolute value of x gives a familiar example at zero. It reaches the point cleanly, yet its direction changes too suddenly for one local slope.
Piecewise rules are especially helpful for testing the difference. A function may use one formula on the left of a joining point and another formula on the right. First check whether the two pieces meet at the same height as the actual plotted point.
If they do not, there is a jump or a hole, so a derivative cannot be assigned there. If they meet, inspect the slopes on both sides. Matching heights are not enough.
The pieces can meet in a pointed roof shape, with different incoming and outgoing slopes. They can even meet with an upright tangent. In that case the slopes grow without bound rather than approaching a finite number.
This distinction appears in motion graphs. Suppose position is plotted against time. A continuous position graph means an object does not suddenly teleport from one place to another.
Its velocity is given by the slope of the position graph. If the position graph has a corner, the model says the object changes velocity instantly. An ideal bouncing ball is often drawn this way.
Its height changes continuously until it reaches the floor, then the graph turns sharply upward. At the instant of impact, one velocity belongs just before the bounce and another belongs just after it.
Real impacts occur over a short time, but the simplified model exposes why derivatives can fail. Since acceleration measures change in velocity, the idealized impact needs extra care when using force equals mass times acceleration.
When working problems, do not rely only on how a graph looks at normal screen size. A steep curve can look like a corner, while a very small gap can be missed. Check the rule, the point value, and behavior from each side.
For formulas made from polynomials, powers with suitable domains, sine, cosine, and exponential functions, differentiability usually holds wherever the formula is defined. Warning signs include absolute values, piecewise definitions, roots, fractions with a zero denominator, and endpoints of a domain. At an endpoint, a one-sided rate may be useful in an application, though many calculus courses reserve the usual derivative for interior points with information from both sides.
Keep the order of reasoning clear. Smooth local behavior guarantees an unbroken graph, but an unbroken graph can still contain sharp features.
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.