Concept 04 / 12Limits, Continuity and Differentiability
Differentiability: Definition, Left and Right Derivatives and Relation to Continuity
The derivative as a limit of difference quotients, left-hand and right-hand derivatives, why differentiability implies continuity but not conversely, corners, cusps and vertical tangents, and testing differentiability of piecewise, modulus and greatest-integer functions.
The derivative of at is
,
if this limit exists and is finite. Geometrically it is the slope of the tangent line, the limiting position of secant lines through ; physically it is an instantaneous rate of change. As with any limit, it has two one-sided versions:
Right-hand derivative: .
Left-hand derivative: .
is differentiable at if and only if both exist, are finite and are equal.
Differentiability implies continuity. If exists, then , so . The contrapositive is the quickest test: a function that is discontinuous at cannot be differentiable there.
The converse is false. A continuous function can fail to be differentiable in three ways, shown in the figure.
Corner: finite but unequal one-sided derivatives. at has and .
Vertical tangent: the difference quotient tends to . at has .
Cusp: one-sided derivatives are infinite with opposite signs. at .
Oscillation: (with value at ) is continuous at , but its difference quotient has no limit. By contrast, is differentiable at with derivative , because its quotient . Each extra factor of buys one more level of smoothness.
Testing differentiability of piecewise functions. At a junction point :
Step 1: check continuity. If it fails, stop: not differentiable.
Step 2: compute the one-sided derivatives. For pieces that are differentiable formulas, equals the limit of the left piece's derivative as , and similarly on the right, provided the function is continuous at . Otherwise use the first-principles definition.
Step 3: compare.
Modulus functions. can fail to be differentiable only where . If has a simple root at (crosses zero with non-zero slope), then has a corner there. If and (a repeated root, like or ), then is differentiable at . So is differentiable at while is not, and is not differentiable only at .
Greatest integer and fractional part. and are discontinuous at integers, hence not differentiable there; elsewhere their derivatives are and . For , check continuity first at the points where is an integer.
Algebra of differentiability. Sums, products, quotients (non-zero denominator) and compositions of differentiable functions are differentiable. But a sum of a differentiable and a non-differentiable function is non-differentiable, while two non-differentiable functions can sum to a differentiable one (). A product can be differentiable even if one factor is not: is differentiable at .
Key Takeaways:
- ; differentiable iff , finite.
- Differentiable continuous; continuous differentiable.
- Failure types: corner, cusp, vertical tangent, oscillation.
- is non-differentiable at simple roots of only.
- (value at ): continuous for , differentiable for .
- Check continuity before computing one-sided derivatives.
- Pitfall: equating the limits of the piece derivatives without checking that the function is continuous at the junction.
Brain-Teaser 1: Find so that is differentiable at .
Solution: Continuity at : , so . Derivatives: the left piece has derivative , the right piece has derivative . So , , and then . With these values the left piece is , which meets at with matching slope : the two graphs are tangent there. Differentiability at a junction is 'same point and same slope'.
Brain-Teaser 2: Where is not differentiable?
Solution: is differentiable everywhere, since behaves like a cube near and its derivative is continuous. The first term has simple roots at and , so it has corners at both. Adding a differentiable function does not remove a corner. So is not differentiable at and only. Treating each modulus term separately, and checking whether the inside has a simple or repeated root, gives the answer without any limits.
Brain-Teaser 3: Let for and . Show that exists but is not continuous at .
Solution: by the squeeze theorem. For , . As , the first term tends to but oscillates between and , so does not exist. Hence is differentiable everywhere, but its derivative is discontinuous at . This is why, at a junction, matching the limits of the piece derivatives is only a shortcut: the true definition of is the limit of the difference quotient.
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