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Theorem 1. The Hausdorff dimension of the graph $\Gamma_f$ of $f$ equals $1$.

Proof. Take a partition of $[0,1]$ by intervals of length $1/n$. Since the function is increasing you can cover the graph by boxes of size $1/n\times (f((k+1)/n)-f(k/n))$, $k=0,1,\ldots,n-1$ located above the intervals of partition. The diameters of each of the boxes is bounded by $1/n+(f((k+1)/n)-f(k/n))$ and the sum of diameters is bounded by $$ \sum_{n=0}^{n-1} \frac{1}{n}+\left(f\left(\frac{k+1}{n}\right)-f\left(\frac{k}{n}\right)\right)=2. $$ Since the diameters of the boxes covering the graph converge to $0$ as $n\to\infty$, it follows that the Hausdorff $\mathcal{H}^1$ measure of the graph is bounded by $2$. $\Box$

Note that the the argument used in the proof shows also that the length of the graph is bounded by $L(\Gamma_f)\leq 2$. The graph of $f$ can be parametrized by a curve $\gamma:[0,1]\to\mathbb{R}^2$, $\gamma(t)=(t,f(t))$.

In fact a stronger result is true:

Theorem 2. $\mathcal{H}^1(\Gamma_f)=L(\Gamma_f)\leq 2$.

This is a consequence of a more general results about curves in metric spaces that we will discuss now.

If $\gamma:[a,b]\to (X,d)$ is a curve in a metric space, then its length is defined by $$ L(\gamma)={\rm Var}(\gamma)= \sup\left\{\sum_{i=1}^{n-1}d(\gamma(t_i),\gamma(t_{i+1}))\right\} $$ where the supremum is over all $n$ and all partitions $a=t_1<\ldots<t_n=b$.

$\gamma$ is said to be rectifiable if $L(\gamma)<\infty$.

Theorem 3. Every rectifiable curve $\gamma:[a,b]\to (X,d)$ can be reparametrized as a $1$-Lipschitz curve.

See Theorem 4.2.1 in [1]. This is so called arc-length parametrization.

Theorem 2 is a straightforward consequence of the following more general result and the fact that $L(\Gamma_f)\leq 2$.

Theorem 4. If $\gamma:[a,b]\to (X,d)$ is an injective rectifiable curve and $\Gamma=f([a,b])$, then $$ \mathcal{H}^1(\Gamma)=L(\gamma). $$

Proof. By Theorem 3 we can assume that $\gamma$ is Lipschitz and in the Lipschitz case Theorem 4 is proved is Theorem 4.4.2 in [1]. $\Box$

[1] L. Ambrosio, P. Tilli, Topics on analysis in metric spaces. Oxford Lecture Series in Mathematics and its Applications, 25. Oxford University Press, Oxford, 2004.

Theorem 1. The Hausdorff dimension of the graph $\Gamma_f$ of $f$ equals $1$.

Proof. Take a partition of $[0,1]$ by intervals of length $1/n$. Since the function is increasing you can cover the graph by boxes of size $1/n\times (f((k+1)/n)-f(k/n))$, $k=0,1,\ldots,n-1$ located above the intervals of partition. The diameters of each of the boxes is bounded by $1/n+(f((k+1)/n)-f(k/n))$ and the sum of diameters is bounded by $$ \sum_{n=0}^{n-1} \frac{1}{n}+\left(f\left(\frac{k+1}{n}\right)-f\left(\frac{k}{n}\right)\right)=2. $$ Since the diameters of the boxes covering the graph converge to $0$ as $n\to\infty$, it follows that the Hausdorff $\mathcal{H}^1$ measure of the graph is bounded by $2$. $\Box$

Note that the the argument used in the proof shows also that the length of the graph is bounded by $L(\Gamma_f)\leq 2$. The graph of $f$ can be parametrized by a curve $\gamma:[0,1]\to\mathbb{R}^2$, $\gamma(t)=(t,f(t))$.

In fact a stronger result is true:

Theorem 2. $\mathcal{H}^1(\Gamma_f)=L(\Gamma_f)\leq 2$.

This is a consequence of a more general results about curves in metric spaces that we will discuss now.

If $\gamma:[a,b]\to (X,d)$ is a curve in a metric space, then its length is defined by $$ L(\gamma)={\rm Var}(\gamma)= \sup\left\{\sum_{i=1}^{n-1}d(\gamma(t_i),\gamma(t_{i+1}))\right\} $$ where the supremum is over all $n$ and all partitions $a=t_1<\ldots<t_n=b$.

$\gamma$ is said to be rectifiable if $L(\gamma)<\infty$.

Theorem 3. Every rectifiable curve $\gamma:[a,b]\to (X,d)$ can be reparametrized as a $1$-Lipschitz curve.

See Theorem 4.2.1 in [1]. This is so called arc-length parametrization.

Theorem 2 is a straightforward consequence of the following more general result and the fact that $L(\Gamma_f)\leq 2$.

Theorem 4. If $\gamma:[a,b]\to (X,d)$ is an injective rectifiable curve and $\Gamma=f([a,b])$, then $$ \mathcal{H}^1(\Gamma)=L(\gamma). $$

Proof. By Theorem 3 we can assume that $\gamma$ is Lipschitz and in the Lipschitz case Theorem 4 is proved is Theorem 4.4.2 in [1]. $\Box$

[1] L. Ambrosio, P. Tilli,   Topics on analysis in metric spaces. Oxford Lecture Series in Mathematics and its Applications, 25. Oxford University Press, Oxford, 2004.

Theorem 1. The Hausdorff dimension of the graph $\Gamma_f$ of $f$ equals $1$.

Proof. Take a partition of $[0,1]$ by intervals of length $1/n$. Since the function is increasing you can cover the graph by boxes of size $1/n\times (f((k+1)/n)-f(k/n))$, $k=0,1,\ldots,n-1$ located above the intervals of partition. The diameters of each of the boxes is bounded by $1/n+(f((k+1)/n)-f(k/n))$ and the sum of diameters is bounded by $$ \sum_{n=0}^{n-1} \frac{1}{n}+\left(f\left(\frac{k+1}{n}\right)-f\left(\frac{k}{n}\right)\right)=2. $$ Since the diameters of the boxes covering the graph converge to $0$ as $n\to\infty$, it follows that the Hausdorff $\mathcal{H}^1$ measure of the graph is bounded by $2$. $\Box$

Note that the the argument used in the proof shows also that the length of the graph is bounded by $L(\Gamma_f)\leq 2$. The graph of $f$ can be parametrized by a curve $\gamma:[0,1]\to\mathbb{R}^2$, $\gamma(t)=(t,f(t))$.

In fact a stronger result is true:

Theorem 2. $\mathcal{H}^1(\Gamma_f)=L(\Gamma_f)\leq 2$.

This is a consequence of a more general results about curves in metric spaces that we will discuss now.

If $\gamma:[a,b]\to (X,d)$ is a curve in a metric space, then its length is defined by $$ L(\gamma)=Var(\gamma)= \sup\left\{\sum_{i=1}^{n-1}d(\gamma(t_i),\gamma(t_{i+1}))\right\} $$ where the supremum is over all $n$ and all partitions $a=t_1<\ldots<t_n=b$.

$\gamma$ is said to be rectifiable if $L(\gamma)<\infty$.

Theorem 3. Every rectifiable curve $\gamma:[a,b]\to (X,d)$ can be reparametrized as a $1$-Lipschitz curve.

See Theorem 4.2.1 in [1]. This is so called arc-length parametrization.

Theorem 2 is a straightforward consequence of the following more general result and the fact that $L(\Gamma_f)\leq 2$.

Theorem 4. If $\gamma:[a,b]\to (X,d)$ is an injective rectifiable curve and $\Gamma=f([a,b])$, then $$ \mathcal{H}^1(\Gamma)=L(\gamma). $$

Proof. By Theorem 3 we can assume that $\gamma$ is Lipschitz and in the Lipschitz case Theorem 4 is proved is Theorem 4.4.2 in [1]. $\Box$

[1] L. Ambrosio, P. Tilli, Topics on analysis in metric spaces. Oxford Lecture Series in Mathematics and its Applications, 25. Oxford University Press, Oxford, 2004.

Theorem 1. The Hausdorff dimension of the graph $\Gamma_f$ of $f$ equals $1$.

Proof. Take a partition of $[0,1]$ by intervals of length $1/n$. Since the function is increasing you can cover the graph by boxes of size $1/n\times (f((k+1)/n)-f(k/n))$, $k=0,1,\ldots,n-1$ located above the intervals of partition. The diameters of each of the boxes is bounded by $1/n+(f((k+1)/n)-f(k/n))$ and the sum of diameters is bounded by $$ \sum_{n=0}^{n-1} \frac{1}{n}+\left(f\left(\frac{k+1}{n}\right)-f\left(\frac{k}{n}\right)\right)=2. $$ Since the diameters of the boxes covering the graph converge to $0$ as $n\to\infty$, it follows that the Hausdorff $\mathcal{H}^1$ measure of the graph is bounded by $2$. $\Box$

Note that the the argument used in the proof shows also that the length of the graph is bounded by $L(\Gamma_f)\leq 2$. The graph of $f$ can be parametrized by a curve $\gamma:[0,1]\to\mathbb{R}^2$, $\gamma(t)=(t,f(t))$.

In fact a stronger result is true:

Theorem 2. $\mathcal{H}^1(\Gamma_f)=L(\Gamma_f)\leq 2$.

This is a consequence of a more general results about curves in metric spaces that we will discuss now.

If $\gamma:[a,b]\to (X,d)$ is a curve in a metric space, then its length is defined by $$ L(\gamma)={\rm Var}(\gamma)= \sup\left\{\sum_{i=1}^{n-1}d(\gamma(t_i),\gamma(t_{i+1}))\right\} $$ where the supremum is over all $n$ and all partitions $a=t_1<\ldots<t_n=b$.

$\gamma$ is said to be rectifiable if $L(\gamma)<\infty$.

Theorem 3. Every rectifiable curve $\gamma:[a,b]\to (X,d)$ can be reparametrized as a $1$-Lipschitz curve.

See Theorem 4.2.1 in [1]. This is so called arc-length parametrization.

Theorem 2 is a straightforward consequence of the following more general result and the fact that $L(\Gamma_f)\leq 2$.

Theorem 4. If $\gamma:[a,b]\to (X,d)$ is an injective rectifiable curve and $\Gamma=f([a,b])$, then $$ \mathcal{H}^1(\Gamma)=L(\gamma). $$

Proof. By Theorem 3 we can assume that $\gamma$ is Lipschitz and in the Lipschitz case Theorem 4 is proved is Theorem 4.4.2 in [1]. $\Box$

[1] L. Ambrosio, P. Tilli, Topics on analysis in metric spaces. Oxford Lecture Series in Mathematics and its Applications, 25. Oxford University Press, Oxford, 2004.

The Hausdorff dimension of the graph of $f$ equals $1$. Take a partition of $[0,1]$ by intervals of length $1/n$. Since the function is increasing you can cover the graph by boxes of size $1/n\times (f((k+1)/n)-f(k/n))$, $k=0,1,\ldots,n-1$ located above the intervals of partition. The diameters of each of the boxes is bounded by $1/n+(f((k+1)/n)-f(k/n))$ and the sum of diameters is bounded by $$ \sum_{n=0}^{n-1} \frac{1}{n}+\left(f\left(\frac{k+1}{n}\right)-f\left(\frac{k}{n}\right)\right)=2. $$ Since the diameters of the boxes covering the graph converge to $0$ as $n\to\infty$, it follows that the $H^1$ measure of the graph is bounded by $2$.

In fact the $H^1$ measure of the graph equals its length. The length of increasing function, as in your question, is always bounded from above by $2$. For the proof that, in a quite general setting, the length equals $H^1$ measure, see the book Topics on analysis in metric spaces by Ambrosio and Tilli. I will provide more detailed references when I am in my office.

Theorem 1. The Hausdorff dimension of the graph $\Gamma_f$ of $f$ equals $1$.

Proof. Take a partition of $[0,1]$ by intervals of length $1/n$. Since the function is increasing you can cover the graph by boxes of size $1/n\times (f((k+1)/n)-f(k/n))$, $k=0,1,\ldots,n-1$ located above the intervals of partition. The diameters of each of the boxes is bounded by $1/n+(f((k+1)/n)-f(k/n))$ and the sum of diameters is bounded by $$ \sum_{n=0}^{n-1} \frac{1}{n}+\left(f\left(\frac{k+1}{n}\right)-f\left(\frac{k}{n}\right)\right)=2. $$ Since the diameters of the boxes covering the graph converge to $0$ as $n\to\infty$, it follows that the Hausdorff $\mathcal{H}^1$ measure of the graph is bounded by $2$. $\Box$

Note that the the argument used in the proof shows also that the length of the graph is bounded by $L(\Gamma_f)\leq 2$. The graph of $f$ can be parametrized by a curve $\gamma:[0,1]\to\mathbb{R}^2$, $\gamma(t)=(t,f(t))$.

In fact a stronger result is true:

Theorem 2. $\mathcal{H}^1(\Gamma_f)=L(\Gamma_f)\leq 2$.

This is a consequence of a more general results about curves in metric spaces that we will discuss now.

If $\gamma:[a,b]\to (X,d)$ is a curve in a metric space, then its length is defined by $$ L(\gamma)=Var(\gamma)= \sup\left\{\sum_{i=1}^{n-1}d(\gamma(t_i),\gamma(t_{i+1}))\right\} $$ where the supremum is over all $n$ and all partitions $a=t_1<\ldots<t_n=b$.

$\gamma$ is said to be rectifiable if $L(\gamma)<\infty$.

Theorem 3. Every rectifiable curve $\gamma:[a,b]\to (X,d)$ can be reparametrized as a $1$-Lipschitz curve.

See Theorem 4.2.1 in [1]. This is so called arc-length parametrization.

Theorem 2 is a straightforward consequence of the following more general result and the fact that $L(\Gamma_f)\leq 2$.

Theorem 4. If $\gamma:[a,b]\to (X,d)$ is an injective rectifiable curve and $\Gamma=f([a,b])$, then $$ \mathcal{H}^1(\Gamma)=L(\gamma). $$

Proof. By Theorem 3 we can assume that $\gamma$ is Lipschitz and in the Lipschitz case Theorem 4 is proved is Theorem 4.4.2 in [1]. $\Box$

[1] L. Ambrosio, P. Tilli, Topics on analysis in metric spaces. Oxford Lecture Series in Mathematics and its Applications, 25. Oxford University Press, Oxford, 2004.