Differentiability and continuity

[Yankel]

Dear all,

The function f(x) is defined below:

\[\left \{ \begin{matrix} 3x^{2} &x\leq 1 \\ ax+b & x>1 \end{matrix} \right.\]

I want to find for which values of a and b the function is differential at x = 1.

The test I was given, is to check the continuity of both f(x) and f'(x). This is fairly easy technically. Checking continuity is only calculating two limits and comparing them.

My question is why this is true. Why the continuity of both f(x) and f'(x) at a point means the function is differential there. I mean, it is known that continuity does not imply differentiability...

Thank you !

 

[HallsofIvy]

Dear all,

The function f(x) is defined below:

\[\left \{ \begin{matrix} 3x^{2} &x\leq 1 \\ ax+b & x>1 \end{matrix} \right.\]

I want to find for which values of a and b the function is differential at x = 1.

First the word is "differentiable", not "differential (which is a noun, not an adjective).

The test I was given, is to check the continuity of both f(x) and f'(x). This is fairly easy technically. Checking continuity is only calculating two limits and comparing them.

My question is why this is true. Why the continuity of both f(x) and f'(x) at a point means the function is differential there. I mean, it is known that continuity does not imply differentiability...

Thank you !

Yes, "continuity" is not "sufficient" for "differentiability" but it is "necessary". That is, if a function is not continuous it cannot be differentiable. Further, you are not really checking continuity of f. The derivative is not necessarily continuous but it must satisfy the "intermediate value property" so the "limit from the right" must equal the
"limit from the left". That is what you are checking.