First, you do NOT take left and right limits for both formulas. The first formula is only valid for x< -3 so you want to take the limit as x goes to -3 from below. the second formula is only valid for so you take the limit as x goes to -3 from above.
The function itself is clearly continuous: the limit from below is (-1/2)(9- 3)= -3 and the limit from above is (3/2)(-3+ 1)= (3/2)(-2)= -3 also. Since that joint value is the value of the function at x= -3, the function is continuous at -3.
You are also incorrect to say that "the derivatives exist but are two different values". Because the two one-sided limits are different, the limit itself does not exist and the function is not differentiable.