Final answer:
To prove that a differentiable function must be continuous, the contrapositive approach is used: if a function is not continuous, then it cannot be differentiable. The contradiction found when assuming a differentiable function is not continuous at a certain point supports the original statement.
Step-by-step explanation:
To show that any differentiable function is continuous, we can use the contrapositive approach. The contrapositive of a statement 'If P then Q' is 'If not Q then not P.' In this case, our original statement is 'If a function y(x) is differentiable, then y(x) is continuous.' Therefore, the contrapositive would be 'If y(x) is not continuous, then y(x) is not differentiable.'
To prove the contrapositive, assume y(x) is not continuous at a point c. This means that the limit of y(x) as x approaches c does not exist or doesn't equal to y(c). Now, if y(x) were differentiable at c, then by definition the first derivative of y(x) with respect to space, dy(x)/dx, would exist at c.
But the existence of this derivative requires that y(x) be continuous at c, which we just assumed it is not. This contradiction implies that y(x) cannot be differentiable at a point where it is not continuous, thus proving our contrapositive statement.
So the original statement that a differentiable function must be continuous is true.