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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 !