Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Compare one-sided difference quotients at a nondifferentiable point.
- Justify the conclusion "f is continuous at zero but not differentiable there" using the stated assumptions.
Before you start
Limits and derivatives.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Is f(x)=|x| differentiable at zero?
Why this math matters
Compare one-sided difference quotients at a nondifferentiable point. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The derivative is a two-sided real derivative.
- The absolute-value function has its standard definition.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Find a continuous function with a corner
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Is f(x)=|x| differentiable at zero?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
f(h)−f(0)=|h|
The difference quotient simplifies without approximation.
Work through the mathematics
|h|/h=1 for h>0 and −1 for h<0
The two directions produce different slopes.
Check the conclusion
f is continuous at zero but not differentiable there
Function values approach zero, while the derivative limit fails to be unique.
The result
f is continuous at zero but not differentiable there
Function values approach zero, while the derivative limit fails to be unique.
Common mistakes to catch
- Continuity does not imply differentiability.
- Assigning a single tangent visually cannot replace checking both sides.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Is f differentiable at x=2?
Show a hint
Near two, |x|=x.
Reveal answer and explanation
Yes, derivative one
The corner only occurs at zero.
Practice 2
Does differentiability imply continuity?
Show a hint
Rewrite f(a+h)−f(a) as h times the difference quotient.
Reveal answer and explanation
Yes
A finite derivative limit multiplied by h→0 forces the function difference to vanish.
Take the idea with you
Explain why an objective with a corner may need tools beyond ordinary stationary-point tests.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Construct an integral from finite measurement strips
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