Sequences of Functions

Prove or supply a counterexample: If $f$ is a nondecreasing real valued function on $[0,1]$, then there is a sequence of continuous functions on $[0,1]$, $(f_{n})$, such that for each $x \in [0,1]$,

Let $f_{n} : \mathbb {R} \rightarrow \mathbb {R}$ be differentiable for each $n = 1,2,\ldots$ with $\left|f_{n}'(x)\right| \leq 1$ for all $n$ and $x$. Assume

for all $x$. Prove that $g : \mathbb {R} \rightarrow \mathbb {R}$ is continuous.

Suppose that $(f_{n})$ is a sequence of nondecreasing functions which map the unit interval into itself. Suppose that

pointwise and that $f$ is a continuous function. Prove that $f_{n}(x) \rightarrow f(x)$ uniformly as $n \rightarrow \infty$, $0 \leq x \leq 1$. Note that the functions $f_{n}$ are not necessarily continuous.

Let $f$ and $f_{n}$, $n = 1,2,\ldots$, be functions from $\mathbb {R}$ to $\mathbb {R}$. Assume that $f_{n}(x_{n}) \rightarrow f(x)$ as $n \rightarrow \infty$ whenever $x_{n} \rightarrow x$. Show that $f$ is continuous. Note: The functions $f_{n}$ are not assumed to be continuous.

Suppose that a sequence of functions $f_{n} : \mathbb {R} \rightarrow \mathbb {R}$ converges uniformly on $\mathbb {R}$ to a function $f : \mathbb {R} \rightarrow \mathbb {R}$, and that $c_{n} = \lim_{x \rightarrow \infty } f_{n}(x)$ exists for each positive integer $n$. Prove that $\lim_{n \rightarrow \infty } c_{n}$ and $\lim_{x \rightarrow \infty } f(x)$ both exist and are equal.

Show that if $f$ is a homeomorphism of $[0,1]$ onto itself, then there is a sequence $(p_{n})$, $n = 1,2,3,\ldots$ of polynomials such that $p_{n} \rightarrow f$ uniformly on $[0,1]$ and each $p_{n}$ is a homeomorphism of $[0,1]$ onto itself.

Let $f_{n} : [0,1] \rightarrow [0,\infty )$ be a continuous function, for $n = 1,2,\ldots$. Suppose that one has

Let $f(x) = \lim_{n \rightarrow \infty } f_{n}(x)$ and $M = \sup_{0 \leq x \leq 1} f(x)$.

Let $f_{1}, f_{2}, \ldots$ be continuous functions on $[0,1]$ satisfying $f_{1} \geq f_{2} \geq \cdots$ and such that $\lim_{n \rightarrow \infty } f_{n}(x) = 0$ for each $x$. Must the sequence $(f_{n})$ converge to $0$ uniformly on $[0,1]$?

Let $k \geq 0$ be an integer and define a sequence of maps

For which values of $k$ does the sequence converge uniformly on $\mathbb {R}$? On every bounded subset of $\mathbb {R}$?

Let $f : [0,1] \rightarrow \mathbb {R}$ be continuous and $k \in \mathbb {N}$. Prove that there is a real polynomial $P(x)$ of degree $\leq k$ which minimizes (for all such polynomials)

Let $(P_{n})$ be a sequence of real polynomials of degree $\leq D$, a fixed integer. Suppose that $P_{n}(x) \rightarrow 0$ pointwise for $0 \leq x \leq 1$. Prove that $P_{n} \rightarrow 0$ uniformly on $[0,1]$.

Let $f$ be a real valued continuous function on a compact interval $[a,b]$. Given $\varepsilon > 0$, show that there is a polynomial $p$ such that $p(a) = f(a)$, $p'(a) = 0$, and $\left|p(x)-f(x)\right| < \varepsilon$ for $x \in [a,b]$.

For each positive integer $n$, define $f_{n} : \mathbb {R} \rightarrow \mathbb {R}$ by $f_{n}(x) = \cos nx$. Prove that the sequence of functions $(f_{n})$ has no uniformly convergent subsequence.

Let $C_{[0,1]}$ denote the space of continuous functions on $[0,1]$. Define

The Arzela--Ascoli Theorem asserts that the sequence $(f_{n})$ of continuous real valued functions on a metric space $\Omega$ is precompact (i.e., has a uniformly convergent subsequence) if

1. $\Omega$ is compact,

2. $\sup \left\| f_{n}\right\| < \infty$ (where $\left\| f_{n}\right\| = \sup \left\{ \left|f_{n}(x)\right| \mid x \in \Omega \right\}$),

3. the sequence is equicontinuous. Give examples of sequences which are not precompact such that: (i) and (ii) hold but (iii) fails; (i) and (iii) hold but (ii) fails; (ii) and (iii) hold but (i) fails. Take $\Omega$ to be a subset of the real line. Sketch the graph of a typical member of the sequence in each case.

Let the functions $f_{n} : [0,1] \rightarrow [0,1]$ ($n = 1,2,\ldots$) satisfy $\left|f_{n}(x)-f_{n}(y)\right| \leq \left|x-y\right|$ whenever $\left|x-y\right| \geq 1/n$. Prove that the sequence $(f_{n})_{n=1}^{\infty }$ has a uniformly convergent subsequence.

Let $(f_{n})$ be a sequence of real valued $C^{1}$ functions on $[0,1]$ such that, for all $n$,

Prove that the sequence has a subsequence that converges uniformly on $[0,1]$.

Let $M$ be the set of real valued continuous functions $f$ on $[0,1]$ such that $f'$ is continuous on $[0,1]$, with the norm

Which subsets of $M$ are compact?

Let $(a_{n})$ be a sequence of nonzero real numbers. Prove that the sequence of functions $f_{n} : \mathbb {R} \rightarrow \mathbb {R}$

has a subsequence converging to a continuous function.

Let $(f_{n})$ be a sequence of continuous functions from $[0,1]$ to $\mathbb {R}$. Suppose that $f_{n}(x) \rightarrow 0$ as $n \rightarrow \infty$ for each $x \in [0,1]$ and also that, for some constant $K$, we have

for all $n$. Does

Let $(g_{n})$ be a sequence of twice differentiable functions on $[0,1]$ such that $g_{n}(0) = g_{n}'(0) = 0$ for all $n$. Suppose also that $\left|g_{n}''(x)\right| \leq 1$ for all $n$ and all $x \in [0,1]$. Prove that there is a subsequence of $(g_{n})$ which converges uniformly on $[0,1]$.

Let $K$ be a continuous real valued function defined on $[0,1] \times [0,1]$. Let $F$ be the family of functions $f$ on $[0,1]$ of the form

with $g$ a real valued continuous function on $[0,1]$ satisfying $\left|g\right| \leq 1$ everywhere. Prove that the family $F$ is equicontinuous.

Let $(g_{n})$ be a sequence of Riemann integrable functions from $[0,1]$ into $\mathbb {R}$ such that $\left|g_{n}(x)\right| \leq 1$ for all $n$, $x$. Define

Prove that a subsequence of $(G_{n})$ converges uniformly.

Let $(f_{n})$ be a sequence of continuous real functions defined $[0,1]$ such that

for all $n$. Define $g_{n} : [0,1] \rightarrow \mathbb {R}$ by

Let $(f_{n})$ be a sequence of continuous functions defined from $[0,1] \rightarrow \mathbb {R}$ such that

Let $K : [0,1] \times [0,1] \rightarrow \mathbb {R}$ be continuous. Define $g_{n} : [0,1] \rightarrow \mathbb {R}$ by

Prove that the sequence $(g_{n})$ converges uniformly.

Let $\varphi_{1}, \varphi_{2}, \ldots , \varphi_{n}, \ldots$ be nonnegative continuous functions on $[0,1]$ such that the limit

exists for every $k = 0,1,\ldots$. Show that the limit

exists for every continuous function $f$ on $[0,1]$.

Let $\lambda_{1}, \lambda_{2}, \ldots , \lambda_{n}, \ldots$ be real numbers. Show that the infinite series

converges uniformly over $\mathbb {R}$ to a continuous limit function $f : \mathbb {R} \rightarrow \mathbb {C}$. Show, further, that the limit

exists.

Define the function $\zeta$ by

Prove that $\zeta (x)$ is defined and has continuous derivatives of all orders in the interval $1 < x < \infty$.

Let $f$ be a continuous real valued function on $\mathbb {R}$. For $n \in \mathbb {N}$, define

Prove that $f_{n}(x)$ converges uniformly to a limit on every finite interval $[a,b]$.

Let $f$ be a continuous real valued function on $\mathbb {R}$ satisfying

where $C$ is a positive constant. Define the function $F$ on $\mathbb {R}$ by

Show that for any continuous function $f : [0,1] \rightarrow \mathbb {R}$ and $\varepsilon > 0$, there is a function of the form

for some $n \in \mathbb {Z}$, where $C_{0}, \ldots , C_{n} \in \mathbb {Q}$ and $\left|g(x)-f(x)\right| < \varepsilon$ for all $x$ in $[0,1]$.