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The key word is "removable". Let's do this.
The function g(h) is discontinuous when h = 0.
The limit of g(h) when h goes to zero is f'(a).
For the right arrow:
We know that the function has a removable discontinuity at h = 0. By the definition of removable discontinuity, the limits when h approaches zero from the left and from the right are the same. That is also the definition of a differentiable function with the formula g(h), and that proves the right arrow.
For the left arrow:
If f'(a) exists, we know that the limit of g(h) when h goes to 0 from the left is the same as the limit of g(h) when h goes to 0 from the right. That's the definition of differentiability: the right and left limits of f'(a) = lim h->0 g(h) are the same. We also know that g(h) is discontinuous when h = 0, but because its limits from the left and from the right when h approaches zero are the same, we know that the discontinuity is removable. This proves the left arrow.