Differentiability and Continuity
Differentiable implies continuous but not the converse.
Key idea
Differentiable implies continuous but not the converse.
In depth
A differentiable function is continuous, because f(x) − f(a) = [(f(x) − f(a))/(x − a)](x − a) → f′(a) · 0 = 0. The converse is false: |x| is continuous but has no derivative at 0.
Worked example
|x| at 0: left slope −1, right slope 1, so no derivative.
Real-world uses
- Explaining why a smooth-looking road can still have a sharp corner that is not differentiable
Common mistake
Believing continuous implies differentiable.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Differentiability and Continuity?
Differentiable implies continuous but not the converse. It is a topic at the expert level in Analysis & Topology.
Where is Differentiability and Continuity used in real life?
Real-world examples: Explaining why a smooth-looking road can still have a sharp corner that is not differentiable.
What should I learn before Differentiability and Continuity?
Make sure you are comfortable with Extreme Value Theorem, the previous topic in Analysis & Topology.
What is a common mistake with Differentiability and Continuity?
Believing continuous implies differentiable.
How can I practise Differentiability and Continuity?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Differentiability and Continuity?
Move on to Weierstrass Function next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.