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I have made the function continuous by equating the RHS and LHS limits and then solving for the variable a.

The limit of the first derivatives of the LHS and RHS are the same, both 0, so that makes the first derivative continuous...

However, the limits of the second derivates for the RHS and LHS are different. One is infinty when the other is 0.

My question is what does this mean exactly? Is the question not possible if my second derivatives are not continuous?