Differentiability and Its Relation to Continuity
f is differentiable at a when the limit of the difference quotient exists there; differentiability implies continuity but not conversely.
What is Differentiability and Its Relation to Continuity?
f is differentiable at a when the limit of the difference quotient exists there; differentiability implies continuity but not conversely.
Key formula / rule: Differentiability requires equal left-hand and right-hand derivatives.
Key points
- Derive the derivative of a function from first principles.
- Analyse differentiability of a piecewise function at a junction.
Common exam trap
Concluding differentiability from continuity alone.
Definitions
- Term
Differentiability and Its Relation to Continuity
- Meaning
f is differentiable at a when the limit of the difference quotient exists there; differentiability implies continuity but not conversely.
- Term
Differentiability and Its Relation to Continuity — explanation
- Meaning
A corner or a vertical tangent breaks differentiability while continuity survives, which is why the modulus function is the standard counterexample. Matching both value and slope is what parameter problems require.
Learning objectives
Derive the derivative of a function from first principles.
Analyse differentiability of a piecewise function at a junction.
Formulae
- Key point
Differentiability requires equal left-hand and right-hand derivatives.
- Key point
Every differentiable function is continuous; |x| shows the converse fails.
- Key point
A vertical tangent means the derivative limit is infinite, so it does not exist.
Prerequisites
MAT-U2-LC-T1-S2-C1
Common mistakes
Concluding differentiability from continuity alone.
Differentiating each piece separately and never comparing them at the junction.
Keywords
Differentiability
Relation
Continuity
Practice preview
Check the differentiability of the function f(x) = 2x + 3 at x = 1.…
easy
Let f(x) = |x - 1| + |x - 2|. At how many points is f(x) not differentiable?…
medium
If a function f(x) is differentiable at a point x = a, then which of the following statements is always true?…
easy