Conditions for Continuity and Differentiability

Question

All of the following statements regarding the continuity and differentiability of piecewise-defined functions are true EXCEPT:

Property | Requirement
:---: | :---:
Continuity | $$\lim_{x	o a^-} f(x) = \lim_{x	o a^+} f(x) = f(a)$$
Differentiability | $$\lim_{x	o a^-} f'(x) = \lim_{x	o a^+} f'(x)$$

Accepted Answer

If a piecewise function is continuous at a boundary point, then it is automatically differentiable at that point.

Answer

Continuity at a boundary point requires that the left-hand and right-hand limits match the value of the function at that point.

Answer

Differentiability requires that the left-hand and right-hand derivatives are equal at the boundary point.

Answer

A function can be continuous at a point without being differentiable if there is a sharp corner or cusp at that point.

Property	Requirement
Continuity	$\lim_{x\to a^-} f(x) = \lim_{x\to a^+} f(x) = f(a)$
Differentiability	$\lim_{x\to a^-} f'(x) = \lim_{x\to a^+} f'(x)$

Conditions for Continuity and Differentiability

Question Leaderboard