数学代写|MATH450 Real Analysis

Problem 1 (10 points): Give the correct definition or statement.
(1) Define the derivative of a function $f$ at a point $x$.
(2) Define local maximum.
(3) State the generalized mean value theorem.
(4) State Taylor’s theorem.
(5) Define partition.
(6) Define a common refinement of two partitions.
(7) Define the Riemann-Stieltjes integral $\int f d \alpha$.
(8) State the fundamental theorem of calculus.
(9) Define a curve in $\mathbb{R}^d$.
(10) Define the length of a curve.

Problem 2 (10 points): Give an example of each, or explain why it can’t exist:
(1) A function $f$ differentiable at 0 , but not continuous at zero.
(2) A continuous function $f$ on the reals that is not differentiable at 0 .
(3) A function $f$ differentiable and continuous at zero, but not continuous anywhere else.
(4) A continuous and differentiable function $f$ on the whole real line so that $f^{\prime}$ is not continuous at 0.
(5) An increasing function that has a negative derivative at 0 .
(6) An increasing, continuous function that is not differentiable at infinitely many points.
(7) A function on $[0,1]$ that is not Riemann integrable.
(8) A function on $[0,1]$ that has infinitely many discontinuities, but is Riemann integrable.
(9) A sequence of Riemann integrable functions that converges at every $x$ to a function that is not Riemann integrable.
(10) A curve in $\mathbb{R}^2$ that is not rectifiable.

Problem 3 (5 points): Give a proof of one the following statements.
(1) If $f$ is increasing and bounded on $[0,1]$ then it is Riemann integrable.
(2) Suppose $f \geq 0$ and is continuous on $[0,1]$. If $\int_1^b f d x=0$ for all $0 \leq a<b \leq 1$, prove that $f(x)=0$ for all $x \in[0,1]$.
(3) Prove that if $\left{f_n\right}$ are Riemann integrable functions on $[0,1]$ that converge uniformly to a function $f$, then $f$ is also Riemann integrable on $[0,1]$.

Problem 4 (5 points): Give a proof of one the following statements.
(1) Prove that there is an infinitely differentiable function $f$ that is zero for $x \leq 0$ and positive for $x>0$.
(2) Prove that for every integer $n>0$ there is a polynomial $p$ of degree $n$ so that
$$
\max _{x \in[0,1]}\left|e^x-p(x)\right| \leq \frac{e}{n !}
$$
(3) Suppose $f$ is infinitely differentiable and $\left|f^{(n)}\right| \leq 1$ for every $n$. Show that if $f$ has infinitely many zeros in a bounded set it must be the constant zero function.