Answer
Statement: If a function is differentiable, then it is continuous.
Contrapositive: If a function is not continuous, then it is not differentiable.
Converse: If a function is continuous, then it is differentiable.
Inverse: If a function is not differentiable, then it is not continuous.
The statement and its contrapositive are logically equivalent. The statement is true, therefore so is its contrapositive.
The converse and inverse are contrapositives of each other so they are logically equivalent. The converse is false. Consider the function f(x) = |x|. f(x) is continuous but it is not differentiable at x=0.
Thus the converse is false and so is the inverse.
Work Step by Step
A statement of the form: $\forall x \in D$, if P(x) then Q(x),
has as its contrapositive statement: $\forall x \in D$, if ~Q(x) then ~P(x),
as its converse statement: $\forall x \in D$, if Q(x) then P(x),
and as its inverse statement: $\forall x \in D$, if ~P(x) then ~Q(x).
