The statement of $g(x)$ is clearly truncated in the prompt, but from the options and the given data

• $f:(a,b)\to[1,\infty)$ is continuous,
• the critical points are $a$ and $b$,

this is the standard piecewise definition

We analyze continuity and differentiability at $a$ and $b$.

1. Continuity inside each region

• For $x<a$ and $x>b$, $g(x)=0$, so $g$ is continuous and differentiable there.
• For $a<x<b$, $g(x)=f(x)$, and since $f$ is continuous on $(a,b)$, $g$ is continuous there.

So only $x=a$ and $x=b$ need checking.

2. Continuity at $x=a$

We have $g(a)=0.$ For continuity at $a$, we need $\lim_{x\to a^-}g(x)=\lim_{x\to a^+}g(x)=g(a).$ Now, $\lim_{x\to a^-}g(x)=0$ since $g(x)=0$ for $x\le a$. Also, $\lim_{x\to a^+}g(x)=\lim_{x\to a^+}f(x).$ But $f(x)\in[1,\infty)$ for all $x\in(a,b)$, so in particular the right-hand limit (if finite) cannot be $0$; it is at least $1$ or may fail to exist, but in any case it is not $0$. Hence $g$ is not continuous at $a$.

Therefore option A ("continuous but not differentiable at $a$") is false.

3. Continuity at $x=b$

Similarly, $g(b)=0.$ For $x\to b^+$, $g(x)=0$, so $\lim_{x\to b^+}g(x)=0.$ For $x\to b^-$, $\lim_{x\to b^-}g(x)=\lim_{x\to b^-}f(x).$ Again, since $f(x)\ge 1$ on $(a,b)$, this left-hand limit cannot be $0$. So $g$ is not continuous at $b$.

Therefore option C ("continuous but not differentiable at $b$") is false.

4. Differentiability at $a$ and $b$

Differentiability implies continuity. Since $g$ is not continuous at either $a$ or $b$, it cannot be differentiable at either point. Thus:

• option B (differentiable on $\mathbb R$) is false,
• option D (continuous and differentiable at either $a$ or $b$ but not both) is also false.

5. Final conclusion

With the natural intended piecewise definition, $g(x)=0\text{ outside }(a,b),\quad g(x)=f(x)\text{ on }(a,b),$ we get:

• $g$ is discontinuous at both $a$ and $b$,
• hence not differentiable at both $a$ and $b$.

So none of the given options is correct.

This does not agree with the stored answer $A,C$.

Most likely, the printed definition of $g(x)$ in the prompt is incomplete/corrupted. Under the standard interpretation above, the stored answer is incorrect.