1. For any integer $k \geq 0$, define a linear functional on $C_0^\infty (\mathbb{R})$:
   $$\langle \delta^{(k)}, \varphi \rangle = (-1)^k \varphi^{(k)}(0),$$
   where $\varphi^{(k)}$ denotes the $k$-th derivative. Prove that this defines a distribution on $\mathbb{R}$ and that its order is precisely $k$.

2. Define a linear functional on $C_0^\infty (\mathbb{R})$:
   $$\langle u, \varphi \rangle = \sum_{k=0}^{\infty} \varphi^{(k)}(0).$$
   Prove that this defines a distribution on $\mathbb{R}$ but we cannot define its order.

3. Prove that the order of $\text{vp} \, \frac{1}{x}$ is 1.

4. Prove that if the test function $\varphi \in C_0^\infty (\mathbb{R})$ satisfies $\varphi(0) = 0$, then
   $$\left\langle \text{vp} \, \frac{1}{x}, \varphi \right\rangle = \int_{\mathbb{R}} \frac{\varphi(x) - \varphi(0)}{x} \, dx.$$

5. Prove that the function on $\mathbb{R}_{>0}$
   $$f(x) = e^{\frac{1}{x}} \mathbf{1}_{(0,+\infty)}(x)$$
   cannot be extended to a distribution on $\mathbb{R}$, i.e., there does not exist $u \in \mathcal{D}'(\mathbb{R})$ such that for any $\varphi \in \mathcal{D}((0, \infty))$,
   $$\langle u, \varphi \rangle = \int_0^{\infty} \frac{1}{x} \varphi(x) \, dx.$$

6. For any $\varepsilon > 0$, define
   $$f_\varepsilon (x) = \frac{1}{x} \mathbf{1}_{(-\infty, -\varepsilon) \cup (\varepsilon, +\infty)}(x).$$
   Prove that as $\varepsilon \to 0$, $f_\varepsilon$ has a limit in the sense of distributions and compute this limit.

7. For any $\varepsilon > 0$, define
   $$g_\varepsilon (x) = \frac{1}{x} \mathbf{1}_{(-\infty, -\varepsilon^2) \cup (\varepsilon^2, +\infty)}(x).$$
   Consider these as a family of distributions in $\mathcal{D}'(\mathbb{R})$. Prove that as $\varepsilon \to 0$, $g_\varepsilon$ does not have a limit in the sense of distributions.

8. Suppose $\mu$ is a probability measure on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$, i.e., $\mu(\mathbb{R}) = 1$. Let $f(x) = \mu((-\infty, x))$, prove that in the sense of distributions
   $$f' = \mathcal{D}'(\mathbb{R}) = \mu.$$

9. Conversely, given a monotonically increasing function $f(x)$ on $\mathbb{R}$ such that
   $$\lim_{x \to -\infty} f(x) = 0, \quad \lim_{x \to +\infty} f(x) = 1,$$
   prove that its derivative in the sense of distributions $f'$ can be viewed as a probability measure on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$, i.e., there exists a probability measure $\mu$ such that for any $\varphi \in \mathcal{D}(\mathbb{R})$,
   $$\langle f', \varphi \rangle = \int_{\mathbb{R}} \varphi \, d\mu.$$

10. For $\alpha \in \mathbb{R}$, define the function on $\mathbb{R}$
    $$x_+^{\alpha} = \begin{cases} x^{\alpha}, & x > 0; \\ 0, & x \leq 0. \end{cases}$$
    Clearly, when $\alpha > -1$, $x_+^{\alpha}$ is locally integrable, thus giving a distribution on $\mathbb{R}$. When $\alpha > 0$, prove
    $$(x_+^{\alpha})' = \alpha (x_+^{\alpha-1}).$$

11. If $\alpha$ is a positive integer, as a distribution on $\mathbb{R}$ (and as the derivative of a distribution), prove
    $$(x_+^{\alpha})^{(\alpha)} = \alpha! H(x).$$
    Here $H(x)$ is the Heaviside function.

12. Now assume $-2 < \alpha < -1$, for any $\varphi \in C_0^\infty (\mathbb{R})$ and any $\varepsilon > 0$, prove
    $$\int_{\varepsilon}^{\infty} x^{\alpha} \varphi(x) \, dx = -(\alpha + 1)^{-1} \varphi(\varepsilon) \varepsilon^{\alpha + 1} - (\alpha + 1)^{-1} \int_{\varepsilon}^{\infty} x^{\alpha + 1} \varphi'(x) \, dx.$$
    This defines the distribution
    $$\langle x_+^{\alpha}, \varphi \rangle = \lim_{\varepsilon \to 0} (\alpha + 1)^{-1} \int_{\varepsilon}^{\infty} x^{\alpha + 1} \varphi'(x) \, dx = (\alpha + 1)^{-1} \langle (x_+^{\alpha + 1})', \varphi \rangle.$$
    For general non-integer $\alpha < -1$, a similar definition can be made, satisfying the relation in Question 10.

13. Find all $u \in \mathcal{D}'(\mathbb{R})$ such that in the sense of distributions
    $$x \cdot u = 0.$$

14. Find all distributions $u$ on $\mathbb{R}$ such that
    $$x \cdot u = \delta_0.$$

15. Prove that for each $u \in \mathcal{D}'(\mathbb{R})$, there exists $\tilde{u} \in \mathcal{D}'(\mathbb{R})$ such that in the sense of distributions
    $$x \cdot \tilde{u} = u.$$

16. Find all distributions $u$ on $\mathbb{R}$ such that
    $$x \cdot u = 1.$$

17. Find all distributions $u$ on $\mathbb{R}$ such that
    $$x \cdot u = \text{pv} \, \frac{1}{x}.$$

18. Find all distributions $u$ on $\mathbb{R}$ such that
    $$x \cdot u = x_+^{\alpha},$$
    where $\alpha < -1$ and is not an integer.

19. Find all $u \in \mathcal{D}'(\mathbb{R})$ such that in the sense of distributions
    $$u + x \cdot u' = 0.$$

20. For a test function $\varphi \in C_0^\infty(\mathbb{R}^n)$ and a real number $\lambda \neq 0$, define
    $$\varphi_{\lambda}(x) = \varphi \left( \frac{x}{\lambda} \right).$$
    For a distribution $u \in \mathcal{D}'(\mathbb{R}^n)$, define the distribution $u_{\lambda}$ as
    $$\langle u_{\lambda}, \varphi \rangle = \langle u, |\lambda|^n \varphi_{\lambda} \rangle.$$
    If for each $\lambda > 0$,
    $$u_{\lambda} \overset{\mathcal{D}'}{=} \lambda^d u,$$
    then $u$ is called a homogeneous distribution of degree $d$, where $d \in \mathbb{R}$. Prove that $\text{vp} \, \frac{1}{x}$ and $x_+^{\alpha}$ (where $\alpha < -1$ and is not an integer) are homogeneous distributions on $\mathcal{D}'(\mathbb{R})$ and determine their degree.

21. Prove that $\delta_0$ is a homogeneous distribution on $\mathcal{D}'(\mathbb{R}^n)$ and determine its degree.

22. Assume $u \in \mathcal{D}'(\mathbb{R}^n)$ is a homogeneous distribution of degree $d$, prove that $\partial_{x_i} u \in \mathcal{D}'(\mathbb{R}^n)$ is a homogeneous distribution of degree $d - 1$.

23. Given $N$ homogeneous distributions $u_1, u_2, \dots, u_N \in \mathcal{D}'(\mathbb{R}^n)$ with distinct degrees, prove they are linearly independent in $\mathcal{D}'(\mathbb{R}^n)$.

24. Given a test function $\varphi_0 \in \mathcal{D}(\mathbb{R}^n)$, a positive real number $\lambda$ and $\lambda_0$, define
    $$\varphi_{\lambda}(x) = \frac{\varphi_{\lambda}(x) - \varphi_{\lambda_0}(x)}{\lambda - \lambda_0}.$$
    Prove that as $\lambda \to \lambda_0$, the function above has a limit in $\mathcal{D}(\mathbb{R}^n)$ and calculate this limit.

25. (Euler’s Formula for Homogeneous Distributions) Given $u \in \mathcal{D}'(\mathbb{R}^n)$, $d \in \mathbb{R}$. Prove that $u$ is a homogeneous distribution of degree $d$ if and only if
    $$\sum_{k=1}^n x_k \frac{\partial u}{\partial x_k} = d \cdot u.$$

26. Find all homogeneous distributions of degree 0 and 1 in $\mathcal{D}'(\mathbb{R}^1)$.