Connecting Differentiability and Continuity

The Big Idea

Continuity and differentiability are two closely related ideas, but they are not the same thing. Continuity asks whether a function has a break, jump, or hole at a point. Differentiability asks whether the function has a well-defined tangent line, or equivalently a finite derivative, at that point.

The key relationship you need to remember for AP Calculus is:

In words: if a function is differentiable at a point, then it must also be continuous at that point.

However, the reverse statement is not true. A function can be continuous at a point and still fail to be differentiable there.

This distinction appears frequently in AP Calculus questions, especially when you are asked to determine whether a function is differentiable at a point from its graph or from a piecewise definition.

What Does Continuity Mean?

A function is continuous at when three conditions are satisfied:

1. exists.
2. exists.
3. .

Informally, continuity means that you can draw the graph through the point without lifting your pencil. There is no hole, jump, or other break at the point.

What Does Differentiability Mean?

A function is differentiable at if its derivative exists at that point. From the definition of the derivative:

This limit represents the slope of the tangent line to the graph at . If the limit exists as a finite number, the function has a well-defined instantaneous rate of change there.

Geometrically, differentiability means the graph is locally smooth enough to have a unique tangent line. A sharp corner, cusp, vertical tangent, or discontinuity can prevent differentiability.

Why Differentiability Implies Continuity

The implication from differentiability to continuity follows directly from the definition of the derivative. Suppose is differentiable at . Then the derivative exists:

Because this limit exists, the change in the function value becomes arbitrarily small as approaches . In other words, the function approaches the actual value .

Therefore:

So a discontinuous function cannot possibly be differentiable at the same point.

Why Continuity Does Not Imply Differentiability

This is the direction students most often get wrong. A function can be perfectly continuous while still having a point where the derivative does not exist.

The classic example is:

This function is continuous at . Its graph passes smoothly through the origin in the sense of having no break, hole, or jump.

But it has a sharp corner at . The slope approaching the origin from the left is different from the slope approaching from the right.

Since the left-hand and right-hand derivatives are not equal, the derivative at zero does not exist.

Therefore, is continuous but not differentiable at .

The Most Important Logical Relationship

Think of differentiability as the stronger condition:

That means there are three useful conclusions you should be able to make immediately:

• If a function is differentiable at a point, it is automatically continuous there.
• If a function is not continuous at a point, it is automatically not differentiable there.
• If a function is continuous at a point, you cannot conclude that it is differentiable there.

A Useful AP Calculus Decision Process

When an AP question asks whether a function is continuous or differentiable at a particular point, use this sequence:

1. Check continuity first. Look for holes, jumps, or other discontinuities.
2. If the function is not continuous, stop. It cannot be differentiable at that point.
3. If the function is continuous, check differentiability. Look for a corner, cusp, vertical tangent, or unequal one-sided derivatives.
4. Remember that continuity alone is not enough.

Graphical Interpretation

On the AP exam, you may be given a graph instead of an equation. The visual clues are extremely useful.

Signs of Discontinuity

• A hole in the graph.
• A jump between the left and right sides.
• A vertical asymptote.
• The function's limiting value does not match its actual value.

Any of these means the function is not continuous at that point, and therefore it is also not differentiable there.

Signs of Nondifferentiability Despite Continuity

• A sharp corner.
• A cusp.
• A vertical tangent.

These can occur even when the graph has no break. That is why you must check differentiability separately from continuity.

One-Sided Derivatives at a Corner

A corner is one of the most important examples of a function that is continuous but not differentiable.

At a point , differentiability requires the left-hand and right-hand derivatives to agree:

If instead

then the derivative does not exist, even if the function itself is continuous.

This is exactly what happens with at the origin.

Continuity vs. Differentiability

A useful way to visualize the logic is:

The set of differentiable functions is contained inside the set of continuous functions. There are continuous functions that are not differentiable, but there are no differentiable functions that are discontinuous at the same point.

Common Mistakes

Practice Problems

The AP Calculus Takeaway

The relationship between continuity and differentiability is one of the most important logical connections in introductory calculus.

When you see a question about differentiability, check continuity first. If continuity fails, differentiability automatically fails. If continuity holds, however, you still need to look for corners, cusps, vertical tangents, or unequal one-sided derivatives.

The simplest memory trick is:

Differentiable → Continuous. But Continuous ↛ Differentiable.