Let f be Riemann integrable on [a,b]. For a ≤ x ≤ b, define
\[F(x) = \int_a^x f(t) dt.\]
Then F is continuous on [a,b]. Furthermore, if f is continuous at a point x_0 in [a,b], then F is differentiable at x_0, and F'(x_0) = f(x_0).
Proof:
Answer (2)
The function F ( x ) = ∫ a x f ( t ) d t is continuous on the interval [ a , b ] . If f is continuous at a point x 0 , then F is differentiable at x 0 , and F ′ ( x 0 ) = f ( x 0 ) . This is supported by the Fundamental Theorem of Calculus.
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To tackle the student’s question, we want to explain why the function F ( x ) = ∫ a x f ( t ) d t is continuous on the interval [ a , b ] , and further, if f is continuous at some point x 0 , then F is differentiable at x 0 and F ′ ( x 0 ) = f ( x 0 ) .
Let's break this down step-by-step:
Continuity of F ( x ) :
For F ( x ) to be continuous on [ a , b ] , it must not have any jumps, breaks, or holes in that interval. Since f is Riemann integrable on [ a , b ] , it means it doesn't have too many discontinuities (only a set of discontinuities of measure zero), and hence the integral can be calculated over this interval.
For any given 0"> e p s i l o n > 0 , we can find a 0"> d e lt a > 0 such that if ∣ x − y ∣ < d e lt a (for x , y ∈ [ a , b ] ), then ( |
\int_x^y f(t) \ dt| < \epsilon ). This shows that F ( x ) changes gradually, ensuring continuity.
Differentiability of F ( x ) at x 0 :
If f is continuous at a point x 0 in [ a , b ] , then to prove that F is differentiable at x 0 , we can use the Fundamental Theorem of Calculus.
According to this theorem, if F ( x ) = ∫ a x f ( t ) d t , and f is continuous at x 0 , then:
F ′ ( x 0 ) = h → 0 lim h F ( x 0 + h ) − F ( x 0 ) = h → 0 lim h 1 ∫ x 0 x 0 + h f ( t ) d t = f ( x 0 ) .
Here, we've used the point-wise continuity of [tex]f[/tex] at [tex]x_0[/tex], ensuring that as [tex]h \to 0[/tex], the average value of [tex]f[/tex] over the interval approaches [tex]f(x_0)[/tex]. Therefore, [tex]F[/tex] is indeed differentiable at [tex]x_0[/tex] and the derivative [tex]F'(x_0) = f(x_0)[/tex].
In summary, the continuity of F on [ a , b ] stems from f being Riemann integrable, while its differentiability at x 0 specifically relies on the continuity of f at x 0 , neatly demonstrated by the Fundamental Theorem of Calculus.
