Statements about derivatives and integrals My professor gave me one example. 
It's given one intervall  $I=\left [ a,b \right ]\subset \mathbb{R}$ and one function $f:I\mapsto \mathbb{R}$. There is also given 8 statements about derivatives and integrals, so here we go: A: f is Riemann integrable.
B: For all $c\in I$ applies that $\lim_{x\mapsto c}f(x)=f(c)$.
C: For all $c\in I$ applies that $f(x)= \sum_{k=0}^{\infty}a_kx^k$.
D: $f$ is for I five times continuously differentiable.
E: $f$ is continues for I.
F: $f(x)=\int_{a}^{x}g(s)ds$ with one continuous function $g:\left [ a,b \right ] \mapsto \mathbb{R}$.
G: $f$ is bounded.
H: $f$ is for $I$ continuously differentiable.
I had to sort by strength. Answer is:
$C \implies D  \implies H \Leftrightarrow F\implies E \Leftrightarrow B \implies A\implies G$
Can somone explain my why is like this?
I understand why everystatment is true, but I don't understand how to sort them by strength. I only understand why $ D  \implies H$ and that's it.
Can someone explain me this?
A: A function as given in $C$ is, among other things, infinitely many times continuously differentiable (the name for the condition in $C$ is analytic, and next after polynomials, they are the nicest functions you can hope to work with). That should tell you why $C \implies D$, but $D$ does not imply $C$. $D$ obviously implies $H$, but $H$ does not imply $D$, since there are functions that may be differentiated once, but not five times (for instance $|x^3|$).
Taking the comment by Dark above into account, if $H$ holds, then you can show that $F$ holds by setting $f'(x) = g(x)$. Conversly, if $F$ holds, you may differentiate the right-hand side of the equaion to obtain $g(x)$, which means that $f'(x)$ exists and is continuous. So $F\Longleftrightarrow H$ (this specific result is more commonly known as the fundamental theorem of calculus).
Of course, if a function is continuously diferentiable, then it is continuous, but the converse does not hold (see $|x|$ for a counterexample). So we get $H \implies E$. $B$ is just an alternative definition of continuous called sequentially continuous, and it is indeed equivalent to the usual definition as long as your spaces are nice enough (any $\Bbb R^n$ for $n \in \Bbb N$ or subspaces thereof are, in this context, nice enough).
A function that is continuous on a closed interval is necessarily Riemann integrable, but the converse is not true, since you could have
$$
f(x) = \cases{0 & if $x < \frac{a + b}{2}$\\1 & if $x \geq \frac{a+b}{2}$}
$$
which gives us $E \implies A$. Lastly, any Riemann integrable function on a closed interval must be bounded, but
$$
f(x) = \cases{0 & if $x$ is rational\\1 & if $x$ is irrational}
$$
shows that there are bounded functions that are not Riemann integrable. This gives $A \implies G$, which finished the chain.
