The statement provided is incorrect, and I will explain why.

The statement claims:

1. $g(x) = \int_a^b f(t) \, dt$, where $f$ is continuous on $[a, b]$, is a function that is:
   • Continuous on $[a, b]$,
   • Differentiable on $(a, b)$,
   • And $g'(x) = 0$.

Analysis:

1. Definition of $g(x)$:

   • $g(x)$ as given in the statement does not depend on $x$, because the integral $\int_a^b f(t) \, dt$ is a definite integral with fixed limits of integration $a$ and $b$.
   • Therefore, $g(x)$ is a constant function, say $C$, as it evaluates to a fixed value for any given continuous $f(t)$ over $[a, b]$.

2. Continuity of $g(x)$:

   • Since $g(x)$ is a constant function, it is trivially continuous on $[a, b]$.

3. Differentiability of $g(x)$:

   • A constant function is differentiable everywhere, and its derivative is zero.

4. The Claim $g'(x) = 0$:

   • This part is correct for a constant function. The derivative of a constant is zero.

5. Error in the Context:

   • The statement gives the impression that $g(x)$ depends on $x$, but it does not. This creates confusion in its claims about continuity, differentiability, and derivatives.

Conclusion:

The statement is misleading because $g(x) = \int_a^b f(t) \, dt$ is a constant function, not a function of $x$. While it is true that a constant function is continuous and differentiable with a derivative of 0, the construction of $g(x)$ as written does not involve $x$. Hence, the discussion about its differentiability and continuity is trivial but does not reflect a proper use of the Fundamental Theorem of Calculus.

If the intention was to use the Fundamental Theorem of Calculus, the function should have been defined as: $g(x) = \int_a^x f(t) \, dt,$ where $x$ is the upper limit of the integral. In this case:

• $g(x)$ is continuous on $[a, b]$,
• Differentiable on $(a, b)$,
• And $g'(x) = f(x)$.

Would you like further clarification on this or related topics?

Follow-up Questions:

1. How does the Fundamental Theorem of Calculus apply to $g(x)$ if $x$ is the upper limit of integration?
2. What is the difference between definite and indefinite integrals in this context?
3. Can you clarify how $g'(x) = 0$ might be misinterpreted in such a statement?
4. Why is the continuity of $f(t)$ critical for $g(x)$'s differentiability?
5. Can we extend this reasoning to improper integrals?

Tip:

Always ensure that a variable in a function definition (like $x$ here) actively influences the integral or formula; otherwise, the derivative claims might become meaningless or trivial.