Problem 1
Let \(x_{1}, x_{2}, \ldots, x_{n}, \ldots\) be the set of all rational points in \([\mathrm{a}, b]\), enumerated in any way, and let \(h_{\mathrm{n}}=1 / 2^{n}\). Prove that the jump function $$ f(x)=\sum_{x_{*}<_{*}} h_{n} $$ is discontinuous at every rational point and continuous at every irrational point
Problem 10
Following van der Waerden, let
$$
\varphi_{0}(x)=\left\\{\begin{array}{ccc}
x & \text { if } & 0
