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Fill in a table of implications for the integral of a function

Let f be a function such that the integral

    \[ A(x) = \int_a^x f(t) \, dt \]

exists for all x in an interval [a,b]. Consider the following statements:

a. f is continuous at c.
b. f is discontinuous at c.
c. f is increasing on (a,b).
d. f'(c) exists.
e. f' is continuous at c.

\alpha. A is continuous at c.
\beta. A is discontinuous at c.
\gamma. A is convex on (a,b).
\delta. A'(c) exists.
\varepsilon. A' is continuous at c.

Make a table to indicate by T whether the statement in row (a)-(e) implies the statement in column (\alpha) - (\varepsilon). Leave the cell in the table blank if there is no implication.


The requested table is as follows:

    \[ \begin{tabular}{c | c c c c c} & $\alpha$ & $\beta$ & $\gamma$ & $\delta$ & $\varepsilon$ \\ \hline a & T& & & T& \\ b & T& & & & \\ c & T& & T& & \\ d & T& & & T& \\ e & T& & & T&T  \end{tabular} \]

We know the first column has all T’s since the function A(x) = \int_a^x f(t) \, dt is continuous for any function f by the first fundamental theorem we know the derivative A'(x) exists at every point in the open interval (a,b). Since differentiability implies continuity (by example 7 on page 163 of Apostol), we then know that A(x) is continuous at any point c \in (a,b).

We know the second column can have no T’s anywhere since, by the same argument as above, the function A(x) is continuous at every point c \in (a,b). Thus, it cannot be discontinuous at any point c \in (a,b).

The statement a,b,d, and e cannot imply that A(x) is convex on (a,b) since they are statements regarding continuity and existence of derivatives. None of these properties have anything to do with convexity. Then, statement (c) does imply statement (\gamma) since we know that a function is convex if its derivative is increasing. Since f is the derivative of A and it is increasing, we have that A is convex.

3 comments

  1. Anonymous says:

    Hi RoRi, shouldn’t d==>theta? With f'(c) exists, f(x) is continuous at c. Since A'(x)=f(x) ==> A'(x) is continuous at x=c.
    Thanks for the wonder effort by putting up the full solutions!

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