Limits of Functions

Let ( $M, d$ ) be a metric space, $A \subset M, f, g: M \rightarrow \mathbb {R}$, and let $a$ be a cluster point of $A$. Also, let $\alpha$ be a real number. If $\lim_{x \rightarrow a} f(x)$ and $\lim_{x \rightarrow a} g(x)$ both exist prove that

Let ( $M, d$ ) be a metric space, $A \subset M$, and $f, g: M \rightarrow \mathbb {R}$ satisfy $f(x)<g(x)$ for all $x \in A$. If $a$ is a cluster point of $A$ and if $\lim_{x \rightarrow a} f(x)$ and $\lim_{x \rightarrow a} g(x)$ both exist to prove that $\lim_{x \rightarrow a} f(x) \leq \lim_{x \rightarrow a} g(x)$.

Let $F: C[0,1] \rightarrow \mathbb {R}$ be defined by $F(f)=f(0)$, and let $g(x)=x^{2}$. Find $\lim_{f \rightarrow g} F(f)$.

Let $d$ be a metric on $C[0,1]$ defined by

(see Exercise 2.1.3). Let $F: C[0,1] \rightarrow \mathbb {R}$ be defined by $F(f)=f(0)$, and let $g(x)=x^{2}$. Find $\lim_{f \rightarrow g} F(f)$.

Let $d$ be a metric on $\mathbb {R}$ defined by

(see Exercise 2.1.8), and let $\rho$ denote the usual distance on $\mathbb {R}, \rho (x, y)=\left|x-y\right|$. Let $f:(\mathbb {R}, d) \rightarrow (\mathbb {R}, \rho )$ be defined by $f(x)=x^{2}$. Find $\lim_{x \rightarrow 2} f(x)$.

Prove Theorem 2.4.5.

Theorem 2.4.5: Let $\left(a_{n}\right)_{n=1}^{\infty }$ be a sequence of real numbers. Then

Let ( $M, d$ ) be a metric space, let $f: M \rightarrow \mathbb {R}$, and let $a \in M$. Prove that $\limsup_{x \rightarrow a} f(x)$ and $\liminf_{x \rightarrow a} f(x)$ always exist either as finite numbers or $\pm \infty$.

Let $\left\{ a_{n}\right\}$ be a sequence of real numbers. Prove that $\lim \sup a_{n}$ equals $\limsup_{x \rightarrow 0} f(x)$, where the function $f:\left\{ 1,1 / 2,1 / 3, \ldots \right\} \rightarrow \mathbb {R}$ is defined by $f(1 / n)=a_{n}$.

Prove that $\lim_{x \rightarrow a} f(x)$ exists if and only if both limit superiors and both limit inferiors (left and right) are all equal.

If $f$ is defined on $[-c, c]$ and $x_{0} \in (-c, c)$, prove the following:

Let ( $M, d$ ) be a metric space, let $f: M \rightarrow \mathbb {R}$, and let $g: M \rightarrow \mathbb {R}$ be defined by $g(x)=\limsup_{t \rightarrow x} f(t)$. Prove that $g$ is upper semicontinuous.

Let $\left\{ A_{n}\right\}$ be an arbitrary collection of subsets of a set $X$. Prove that

Let ( $M, d$ ) be a metric space and for every $n \in \mathbb {N}$ let $f_{n}: M \rightarrow [0,+\infty ]$. Prove that

Let $f$ and $g$ be real-valued functions on an interval $(a-\eta , a+\eta )$. Prove that

Let $f$ and $g$ be strictly positive functions on an interval $(a-\eta , a+\eta )$ and suppose that $\liminf_{x \rightarrow a} f(x)$ and $\limsup_{x \rightarrow a} f(x)$ exist and are not 0 . Prove that:

Let $f$ and $g$ be nonnegative functions on an interval ( $a-\eta , a+ \eta )$ and suppose that $\liminf_{x \rightarrow a} f(x), \limsup_{x \rightarrow a} f(x), \liminf_{x \rightarrow a} g(x)$, and $\limsup_{x \rightarrow a} g(x)$ exist and are not 0 . Prove that

Let $f$ and $g$ be nonnegative functions on an interval ( $a-\eta , a+\eta$ ) and suppose that $\lim_{x \rightarrow a} f(x)$ exists. Prove that:

Let ( $M, d$ ) be a metric space, $A \subset M$, and $f: M \rightarrow \mathbb {R}$. If $a$ is a cluster point of $A$ prove that:

Prove Corollary 2.4.9.

Corollary 2.4.9: Let $(M,d)$ be a metric space, let $f: M \rightarrow \mathbb {R}$, and let $A \subset M$. If $a$ is a cluster point of $A$ then

where $W(f, a) = \left\{ w : \exists \left(a_{n}\right) \subset A \setminus \left\{ a\right\} , \lim a_{n} = a, \lim f(a_{n}) = w\right\}$ is the set of all subsequential limits of $f$ along sequences in $A \setminus \left\{ a\right\}$ converging to $a$.

Let ( $M, d$ ) be a metric space, let $f: M \rightarrow \mathbb {R}$, and let $A \subset M$. If $a$ is a cluster point of $A$ prove that

Let $a \in \mathbb {R}$ and $f:(a,+\infty ) \rightarrow \mathbb {R}$. Prove that the limit $\lim_{x \rightarrow +\infty } f(x)$ exists if and only if $\limsup_{x \rightarrow +\infty } f(x)=\liminf_{x \rightarrow +\infty } f(x)$ and in this situation they are all equal.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$. Prove that

Let $a \in \mathbb {R}$ and $f:(a,+\infty ) \rightarrow \mathbb {R}$. Prove that $\limsup_{x \rightarrow +\infty } f(x)=L$ if and only if, for any sequence $\left\{ a_{n}\right\}$ converging to $+\infty , \limsup_{x \rightarrow +\infty } f\left(a_{n}\right)=L$.

Let $f:(a,+\infty ) \rightarrow \mathbb {R}$. Prove that

Prove Theorem 2.4.20.

Theorem 2.4.20: Let $f$ be a differentiable function on a set $E \subset \mathbb {R}$. Then $f$ is Lipschitz with constant $K = \sup \left\{ \left|f'(x)\right| : x \in E\right\}$ if and only if it has a bounded derivative.

Suppose that $f$ is a real-valued Lipschitz function on $(a, b)$. Prove that $f$ is uniformly continuous on $(a, b)$.

Let ( $M, d$ ) be a metric space and $x_{0} \in M$. Let $f: M \rightarrow \mathbb {R}$ be defined by $f(x)=d\left(x, x_{0}\right)$. Prove that $f$ is uniformly continuous.

Give an example of a function that is continuous but not uniformly continuous. Give an example of a function that is uniformly continuous but not Lipschitz.

Give an example of a function that is Lipschitz on a set $E$ but not differentiable at a point $x \in E$.

Let $f$ be a real-valued function defined on a set $A \subset \mathbb {R}^{2}$ such that it is a continuous function of $x$ (with $y$ fixed), and let $f$ satisfy the Lipschitz condition with respect to $y$, i.e., suppose that there exists $M>0$ such that

for all $\left(x, y_{1}\right),\left(x, y_{2}\right) \in A$. Prove that $f$ is continuous on $A$.

Let $d(x, y)=\sqrt{\left|x-y\right|}$, let $\rho (x, y)=\left|x-y\right|$, and consider $f$ : $([0,1], d) \rightarrow ([0,1], \rho )$ defined by $f(x)=x^{\alpha }$. Determine what values of $\alpha >0$ is $f$ Lipschitz.

Let $(C[0,1], d)$ be the metric space with $d(f, g)=\sup \left\{ \left|f(x)-g(x)\right|: x \in [0,1]\right\}$, and let $F: C[0,1] \rightarrow C[0,1]$ be defined by $F(f)=f^{2}$. Determine whether $F$ is Lipschitz.

Let ( $C[0,1], d$ ) be the metric space with $d(f, g)=\sup \left\{ \left|f(x)-g(x)\right|: x \in [0,1]\right\}$, and let $F: C[0,1] \rightarrow C[0,1]$ be defined by $F(f)=\sin f$. Determine whether $F$ is Lipschitz.

Let $d: \mathbb {R} \rightarrow \mathbb {R}$ be defined by $d(x, y)=1$ if $x \neq y$ and $d(x, y)=0$ if $x=y$, let $\rho (x, y)=\left|x-y\right|$, and consider $f:(\mathbb {R}, d) \rightarrow (\mathbb {R}, \rho )$ defined by $f(x)=x^{2}$. Determine if $f$ is uniformly continuous. Is it Lipschitz?