Question about fundamental thm calc part 1

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The discussion centers on the differentiability of the function g defined by g(x) = ∫f(t)dt over the interval [a,b], where f is continuous on [a,b]. It is established that g is continuous on [a,b] and differentiable on the open interval (a,b), with the derivative g' equating to f. The key point raised is that differentiability is typically defined on open intervals, which explains why g is not differentiable at the endpoints of the closed interval [a,b]. A definition for differentiability on closed intervals is necessary for g to be considered differentiable across the entire interval.

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If f is continuous on [a,b] then the function g defined by
g(x) = ∫f(t)dt a <= x <= b is continuous on [a,b] and differentiable on (a,b) and g' = f

Question...

If f is continuous on [a,b], then why is g only differentiable on (a,b)?
This does not make sence... if g' = f g should be diff on [a,b] after making the first statement "f is continuous on [a,b]"
 
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The thing is that differentiability is often only defined on open intervals. This is likely what your text has done.

If you want g to be differentiable on [a,b], then you need to have a definition for differentiability on closed intervals.

Mod note: as this is not a homework problem, I mover it to the math forums.
 

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