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Differentiability & Continuity

A derivative is a slope — but not every function has one at every point. This lesson pins down exactly when a derivative exists, the four classic ways it can fail, and the one-sided test the AP exam uses to settle it at the seam of a piecewise function.

Here’s the key relationship: differentiable always implies continuous — if a curve has a slope at a point, it can’t have a break there. But the reverse fails: a curve can be perfectly connected and still have no slope at a point — think of a sharp corner, where the curve abruptly changes direction. Differentiability is the stronger condition: it demands not just connection, but smoothness.

Here is a curve with no break anywhere — you could draw it without lifting your pen. Yet at the marked point, it has no slope at all. How can a perfectly connected curve fail to have a slope?

the curve is continuous here — but watch the slope

The answer is the whole lesson: continuity (no breaks) and differentiability (a well-defined slope) are different things. Connection isn't enough — the curve also has to be smooth.

Differentiable is a one-way street to continuous

If a function has a derivative at a point, it must be continuous there — a slope can't exist across a gap or a jump. So differentiability is the stronger property: every differentiable function is continuous, but plenty of continuous functions fail to be differentiable. That arrow only points one way.

What you're looking at: the logic in both directions. The forward arrow always holds; the backward arrow is crossed out because continuity alone is not enough.

The contrapositive (your fastest tool)

Flip “differentiable \(\Rightarrow\) continuous” around: not continuous \(\Rightarrow\) not differentiable. So the instant you spot a jump, hole, or asymptote, you're done — no derivative can live there.

The interesting cases are the ones that are continuous but still fail. That's the next section.

So the question becomes: if a connected curve can fail to have a slope, exactly when does it happen? It turns out there are only four ways.

The four ways a derivative dies

When a continuous function has no derivative at a point, it's one of four culprits. Tap each to see the graph and the reason. The first three are continuous — the curve is connected, but it isn't smooth.

What you're looking at: the live graph for the selected failure mode, with the trouble point marked.

Corner

continuous here?yes

Corner, cusp, vertical tangent, jump — four different-looking failures. But is there one idea underneath them all? There is, and you can see it by looking up close.

Smooth means “locally straight”

Here's the deepest way to see it. A function is differentiable at a point exactly when, if you zoom in far enough, the curve becomes indistinguishable from a straight line — the tangent. Zoom in on a smooth point and it flattens to a line. Zoom in on a corner and it stays a corner forever.

move the lens

What you're looking at: the full curve, with a magnifier lens you slide along it. Inside the lens you see the curve up close. On the smooth curve, every spot looks like a straight line through the lens. Switch to the corner and slide the lens over the kink — it stays sharp, no matter how close you look.

Local linearity

“Has a derivative” and “looks like a line up close” are the same statement. Whatever the lens reveals as a straight line, its slope is the derivative there.

lens magnification18×

slope under the lens—

looks like a line?—

That zoom-in test is the intuition. Now here’s the version the AP exam actually grades — turning “is it smooth?” into two concrete checks you can compute.

The seam test for piecewise functions

The AP exam's favorite version of this: a function defined in pieces, glued at a seam. To check differentiability there, run two tests in order. First continuous? (do the pieces meet?) Then smooth? (do the one-sided slopes match?). It must pass both.

right-piece slope m right-piece shift b

What you're looking at: left piece is \(f(x)=x^2\) for \(x\le 1\); right piece is the line \(mx+b\) for \(x \gt 1\). Tune \(m\) and \(b\) and watch the two tests.

Two tests at the seam x = 1

left value vs right value

continuous?

left slope vs right slope

slopes match?

differentiable at x = 1?

You can compute it at a seam. But often the exam just hands you a picture and asks you to spot the trouble. Same four culprits — now find them by eye.

Reading it off a graph

On the exam you're often just shown a graph and asked where \(f\) is not differentiable. Scan for the four culprits: breaks (not continuous), corners, cusps, and vertical tangents. Here's a graph with several — can you spot them before revealing?

What you're looking at: a single continuous-looking graph with four trouble points. Each is a different failure mode.

Where f' fails here

Everywhere else, the graph is a smooth curve — differentiable. The derivative exists at every point except these four.

The whole idea on one card

Differentiable at a point always forces continuous there — the forward implication never fails

But continuous does NOT force differentiable: the curve can be connected yet not smooth

Four failure modes: corner, cusp, vertical tangent (all continuous), and discontinuity (not even continuous)

At a piecewise seam, differentiable means the one-sided derivatives exist and are equal

The ideas everything else builds on

One-way implication

Differentiable always implies continuous — a slope can’t exist across a break. This direction never fails.

The reverse fails

Continuous does not imply differentiable. \(|x|\) is continuous at 0 but has a corner — no slope there.

Match the one-sided slopes

A derivative exists at a point only when the left and right slopes agree. At a corner they differ; at a smooth point they match.

The contrapositive shortcut

Not continuous \(\Rightarrow\) not differentiable. Spot a jump, hole, or asymptote and you’re instantly done — no derivative there.

Your turn

Five problems — the implication, the failure modes, and the piecewise seam test (including a solve-for-\(k\)). Try each before revealing.

Quick quiz

Eight fast checks across the whole lesson.

Question 1 of 8

Score: 0 / 0

Common mistakes & exam tips

Assuming continuous means differentiable

The single most common slip. Continuity is necessary but not sufficient — a continuous corner like \(|x|\) has no derivative. Always check smoothness separately.

Getting the implication backwards

It's differentiable \(\Rightarrow\) continuous, not the reverse. Memorize the one-way arrow; the backward direction is false.

Only checking continuity at a seam

On a piecewise problem, matching the values makes it continuous — but you still must match the one-sided slopes for differentiability. Two tests, not one.

Forgetting the vertical-tangent case

\(\sqrt[3]{x}\) is continuous and even has a tangent line at 0 — but it's vertical, so the slope is infinite and \(f'(0)\) doesn't exist.

Solving for k with only one equation

“Make it differentiable” usually needs two conditions (continuity and matching slopes), which may give two unknowns. Set up both equations.

Calling a cusp a corner

A corner has two different finite slopes; a cusp has slopes running to \(\pm\infty\). Both kill the derivative, but name them correctly if asked.

On the AP exam

To justify \(f'(a)\) doesn't exist, cite the specific reason: corner (one-sided slopes differ), cusp/vertical tangent (slope infinite), or discontinuity (not continuous).
For a piecewise “is it differentiable” question, show both the value check and the one-sided derivative check at the seam.
If asked whether \(f\) is continuous given that it's differentiable, the answer is immediately yes — no work needed.
One-sided derivatives \(f'_-(a)\) and \(f'_+(a)\) are the rigorous tool; use them whenever the formula changes at \(a\).

You can now see limits. Next: computing them — the limit laws, why plugging in usually just works, and the algebra toolkit (factoring, conjugates) for when it doesn't, including that \(\tfrac{\sqrt{x+4}-2}{x}\) mystery from the table. Computing Limits is next — or head back to the Unit 2 Guide.