Solution

For the sum

F = f + g

In case (a), **yes**. If $F$ were differentiable at $a$ , then $g = F - f$ would be differentiable there, contrary to the hypothesis.
In case (b), **no**. Take
$f (x) = ∣ x - a ∣, g (x) = - ∣ x - a ∣.$
(22)
Both are continuous and nondifferentiable at $a$ , but $F = 0$ is differentiable.

For the product

G = f g

, the answer is **no** in both cases:

For case (a), take $f (x) = x - a$ and $g (x) = ∣ x - a ∣$ . Then $f$ is differentiable at $a$ , $g$ is not, but
$G (x) = (x - a) ∣ x - a ∣$
(23)
has derivative $0$ at $a$ .
For case (b), take $f (x) = g (x) = ∣ x - a ∣$ . Neither factor is differentiable at $a$ , whereas
$G (x) = (x - a)^{2}$
(24)
is differentiable.

Every displayed

f

and

g

is a continuous function and is nonzero at points arbitrarily close to

a

, so the extra condition in the question is satisfied.

Solved by gpt-5.6-sol high.

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