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Grade 12 Calculus · Lesson Extension75 min

Piecewise Differentiability · Check the Join

At a breaking point, differentiability is a two-gate test: the pieces must meet first, then their slopes must match.

Use Mr. Azzam’s breaking-point routine to connect continuity, one-sided derivatives, and differentiability with interactive decisions.

Continuity firstOne-sided derivativesCorners & cuspsVertical tangentsParameters

Key takeaway

Mr. Azzam’s two-gate test

Gate 1 · Do the pieces meet?

Check continuity at the breaking point. If continuity fails, stop.

Gate 2 · Do the slopes match?

Compare the left-hand and right-hand derivatives. Equal finite slopes mean differentiable.

What does differentiable mean?

Definition

Differentiable at x = c

A function is differentiable at $x=c$ when its derivative exists there. At a piecewise breaking point, we must inspect the behavior from both sides.

f'_{-}(c)=f'_{+}(c)\quad\Longrightarrow\quad f'(c)\text{ exists}

Key takeawayThe relationship you must remember

\text{Differentiable at }c\;\Longrightarrow\;\text{Continuous at }c

\text{Continuous at }c\;\not\Longrightarrow\;\text{Differentiable at }c

Common mistake. Do not say “continuous, therefore differentiable.” A corner can be continuous but still have different left and right slopes.

Key takeawayLogic check

Differentiability is stronger than continuity: differentiable implies continuous, but a continuous graph can still fail differentiability at a corner, cusp, or vertical tangent.

Your turn

Question 1 of 2

Before checking the derivatives at a breaking point

x=c

, what should you check first?

Continuity at

x=c

The second derivativeThe

y

-intercept

Piecewise functions: focus on the breaking point

Away from a breaking point, differentiate the active formula normally. The special work happens where the formula changes.

First look

f(x)=\begin{cases}x^2,&x<1\\2x-1,&x\ge1\end{cases}

The breaking point is $x=1$ . That is the point where we perform the full differentiability check.

Key takeawayWhere should you spend your work?

For a piecewise function, ordinary points use the derivative of the active piece. The joining point is where continuity and both one-sided derivatives must be checked.

Your turn

x=c

, which condition confirms differentiability after continuity is established and both one-sided derivatives are finite?

f'_{-}(c)=f'_+(c)

f(c)=0

f'_{-}(c)=-f'_+(c)

Mr. Azzam’s routine at a breaking point

STEP 1

Identify the breaking point

x=c

STEP 2

Check continuity first

f(c),\;\lim_{x\to c^-}f(x),\;\lim_{x\to c^+}f(x)

STEP 3

Differentiate each piece

Find the derivative formula on each side.

STEP 4

Use the correct side

f'_{-}(c)

uses the left piece;

f'_{+}(c)

uses the right piece.

STEP 5

Compare the slopes

f'_{-}(c)\stackrel{?}{=}f'_{+}(c)

STEP 6

State the conclusion

Differentiable or not differentiable at the breaking point.

Common mistake. If continuity fails, stop. The function cannot be differentiable at that point.

ExplanationParameter problems · use two equations

When constants are chosen to make a piecewise function differentiable, continuity gives the first condition and matching one-sided derivatives gives the second. Solve both conditions together.

Your turn

For

r(x)=\begin{cases}x,&x<0\\x+1,&x\ge0\end{cases}

, both separate formulas have derivative

1

. Is

r

differentiable at

0

Yes, because both formulas have slope

1

No, because the pieces do not meetYes, because

r(0)

existsNo, because the slopes of the formulas are unequal

Worked examples · test the join

Worked example 1

Example 1 · Both gates pass

f(x)=\begin{cases}x^2+1,&x<2\\4x-3,&x\ge2\end{cases}

\lim_{x\to2^-}f(x)=5=f(2)=\lim_{x\to2^+}f(x)

f^{\prime}_{-}(2)=4,\qquad f^{\prime}_{+}(2)=4

Both continuity and matching slopes pass, so the function is differentiable at x=2.

Worked example 2

Example 2 · Continuous, but slopes differ

g(x)=\begin{cases}x+2,&x<1\\x^2+2,&x\ge1\end{cases}

\lim_{x\to1^-}g(x)=3=g(1)=\lim_{x\to1^+}g(x)

g^{\prime}_{-}(1)=1,\qquad g^{\prime}_{+}(1)=2

The function is continuous at x=1, but it is not differentiable there.

Worked example 3

Example 3 · Stop when continuity fails

h(x)=\begin{cases}2x+1,&x<3\\x^2-1,&x\ge3\end{cases}

\lim_{x\to3^-}h(x)=7,\qquad h(3)=8

Continuity fails at x=3, so there is no need to compare derivatives.

Worked example 4

Example 4 · Find two parameters

p(x)=\begin{cases}ax+b,&x<1\\x^2+1,&x\ge1\end{cases}

a+b=2

a=2\quad\Longrightarrow\quad b=0

The function is differentiable at x=1 when a=2 and b=0.

Worked example 5

Example 5 · No parameter satisfies both gates

q(x)=\begin{cases}ax+1,&x<2\\x^2-1,&x\ge2\end{cases}

2a+1=3\Rightarrow a=1

q^{\prime}_{-}(2)=1,\qquad q^{\prime}_{+}(2)=4

The value that gives continuity does not match the slopes, so no value of a makes the function differentiable at x=2.

Your turn

Question 1 of 2

In Example 4,

p(x)=\begin{cases}ax+b,&x<1\\x^2+1,&x\ge1\end{cases}

. Which pair makes

p

differentiable at

1

a=1,\ b=1

a=0,\ b=2

a=2,\ b=2

a=2,\ b=0

Your turn · decide step by step

Interactive lab

Breaking-point investigation

Check continuity first, then compare the two one-sided derivatives.

f(x)=\begin{cases}x^2,&x<1\\2x-1,&x\ge1\end{cases}

Investigation step 1 of 4

A matching value is necessary, but not sufficient. You must also compare slopes.

Step 1 · At the breaking point

x=1

, what must we check first?

When differentiability fails

Jump discontinuity

ExplanationDiscontinuity

A jump, hole, or other discontinuity automatically means the function is not differentiable there.

Corner / sharp turn

ExplanationCorner / sharp turn

The function is continuous, but the one-sided derivatives are different.

|x|:\qquad f'_{-}(0)=-1,\qquad f'_{+}(0)=1

Cusp

ExplanationCusp

The one-sided slopes become unbounded in opposite directions, so the derivative does not exist.

Vertical tangent

ExplanationVertical tangent

The graph is continuous, but the slope becomes infinite. Therefore the ordinary finite derivative does not exist at that point.

Your turn

Question 1 of 2

A continuous graph has finite one-sided derivatives

-2

and

5

c

. What is

f'(c)

Illustrative graph with the stated one-sided slopes, placing c at 0.

3

-10

It does not exist

3/2

Find your next lesson

Piecewise Differentiability · Check the Join

What does differentiable mean?

Your turn

Your turn · student questions

Piecewise functions: focus on the breaking point

Your turn

Your turn · student questions

Mr. Azzam’s routine at a breaking point

Your turn

Your turn · student questions

Worked examples · test the join

Example 1 · Both gates pass

Example 2 · Continuous, but slopes differ

Example 3 · Stop when continuity fails

Example 4 · Find two parameters

Example 5 · No parameter satisfies both gates

Your turn

Your turn · student questions

Your turn · decide step by step

Breaking-point investigation

When differentiability fails

Your turn

Your turn · student questions

Keep your unfinished answers?