Fundamental theorem of calculus

[PFuser1232]

"If $f$ is continuous on $[a,b]$ and:
$$g(x) = \int_a^x f(t) dt$$
Then $g$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $g'(x) = f(x)$."

This is the first fundamental theorem of calculus. I'm curious as to why $g$ is only differentiable on $(a,b)$, but not $[a,b]$.

 

[micromass]

Because you likely have not defined differentiability on closed intervals, only on open ones.

 

[Svein]

This is the first fundamental theorem of calculus. I'm curious as to why g is only differentiable on (a,b), but not [a,b].

Definition. Take point b. In order for the derivative to exist at b, the two limits $\lim_{h\rightarrow 0}\frac{g(b-h)-g(b)}{h}$ and $\lim_{h\rightarrow 0}\frac{g(b+h)-g(b)}{h}$ must both exist and be equal. But $g(b+h)$ is not defined and may not even exist!
The same argument applies to point a (only here it is g(a-h) that is not defined).

 

[PFuser1232]

Definition. Take point b. In order for the derivative to exist at b, the two limits $\lim_{h\rightarrow 0}\frac{g(b-h)-g(b)}{h}$ and $\lim_{h\rightarrow 0}\frac{g(b+h)-g(b)}{h}$ must both exist and be equal. But $g(b+h)$ is not defined and may not even exist!
The same argument applies to point a (only here it is g(a-h) that is not defined).

But can't we say the same about the continuity of $f$ and/or $g$?
In order for $f$ to be continuous at $a$ (implied by the statement "$f$ is continuous on $[a,b]$"), the following must hold true:
$$\lim_{x \rightarrow a} f(x) = f(a)$$
But $\lim_{x \rightarrow a} f(x)$ can only exist if $\lim_{x \rightarrow a^+} f(x) = \lim_{x \rightarrow a^-} f(x)$.

 

[HallsofIvy]

No. What is implied by the statement "f is continuous on [a, b]" is that f is continuous from the right so the only thing that is certain is that $\lim_{x\to a^+} f(x)= f(a)$.

 

[PFuser1232]

No. What is implied by the statement "f is continuous on [a, b]" is that f is continuous from the right so the only thing that is certain is that $\lim_{x\to a^+} f(x)= f(a)$.

But again, why can't the same be said of the differentiability of a function on a closed interval $[a,b]$?

 

[micromass]

Because you haven't defined differentiability on closed intervals, only on open ones. You can define it on closed intervals and then the fundamental theorem holds on those closed intervals, but that is usually not done because it is annoying (and not very useful).

 

[Stephen Tashi]

No. What is implied by the statement "f is continuous on [a, b]" is that f is continuous from the right so the only thing that is certain is that $\lim_{x\to a^+} f(x)= f(a)$.

We need to straighten out whether this is an implication that follows simply from the ordinary definitions of the limit of a function and the definition of a function being continuous at a point or whether it comes from a special definition for the phrase "f(x) is continuous on [a,b]". For example, if a function f(x) does not exist outside the interval [a,b] then does the ordinary definition of "f(x) is continuous at x = c" imply that f(x) is NOT continuous at x = a? Or do the ordinary definitions imply $lim_{x\rightarrow a^+} f(x) = f(a)$ without any need for making a special definition for the phrase "f(x) is continuous on [a,b]" ?

( I think this is related to the controversies one sees in discussions about the meaning of the phrase "f(x) is everywhere continuous" when applied to functions whose domain is a proper subset of the real numbers. In a sophisticated treatment, the continuity of a function depends on the topology being used for the "spaces" of the domain and range. )

 

[PFuser1232]

What if the integrand, $f$, were continuous over $(a,b)$ instead? Would it make any difference?
And what if $f$ were defined for all $x$ in $(a,b)$, but not at $a$ and/or $b$?

 

[Svein]

What if the integrand, f, were continuous over (a,b) instead? Would it make any difference?

No. The integral has the same value (as long as we are talking about relatively standard integration).

And what if f were defined for all x in (a,b), but not at a and/or b?

Same thing.

 

[PFuser1232]

No. The integral has the same value (as long as we are talking about relatively standard integration).
Same thing.

But why is this not mentioned in the fundamental theorem of calculus? Or am I missing something?
I know continuity on $[a,b]$ implies continuity on $(a,b)$ (not the other way around), but does that mean that based on the fundamental theorem (as stated above) I can deduce that continuity on $(a,b)$ doesn't make a difference?

 

[Svein]

Observations: $\int_{a}^{b}f(x)dx=\int_{a}^{a}f(x)dx+\int_{(a, b)}f(x)dx+\int_{b}^{b}f(x)dx$. Since f is continuous over [a, b], f(a) and f(b) exist and are finite. Therefore $\int_{a}^{a}f(x)dx = f(a)\cdot (a - a)=0$. We also have $\int_{b}^{b}f(x)dx = f(b)\cdot (b - b)=0$. Therefore $\int_{a}^{b}f(x)dx=\int_{(a, b)}f(x)dx$

 

[Stephen Tashi]

I'm curious as to why $g$ is only differentiable on $(a,b)$, but not $[a,b]$.

Try $f(x) = 1 - x$ if $x \le 1$ and $f(x) = x - 1$ otherwise and let $a = 0,\ b = 1$. Let $g(x) = \int_0^x f(x) dx$.