# Questions tagged [step-function]

A step function, also known as a simple function, is a finite sum of characteristic functions of bounded intervals. They are often used in measure theory to approximate integrable functions.

149 questions
Filter by
Sorted by
Tagged with
28 views

### Integral of Heaviside($R^2-x^2-y^2$)

I have this integral (where $R$ is a positive constant) $$\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty} H(R^2-x^2-y^2)dxdy$$ and I'm pretty lost when trying to calculate it. I don't ...
35 views

33 views

18 views

46 views

28 views

70 views

### Proving that sin$(1/x)$ is not regulated

I have seen in this question howto prove whether sin$(1/x)$ is not regulated. But i'm not quite sure why it's correct. Since the definition of a regulated function is as follows: This means that the ...
64 views

21 views

### Step functions : $\phi \leq \psi$ implies $\phi_i \leq \psi_i$?

In this question he has stated that $\phi \leq \psi$ implies $\phi_i \leq \psi_i$ for $i \in [1,k]$ I don't think this is correct as there may be a value $\psi_i$ which is smaller than $\phi_i$ ? Or ...
38 views

### Step functions: Using $-|\phi| \leq \phi \leq |\phi|$ conclude $|\int^b_a \phi(x)dx| \leq \int^b_a|\phi(x)|dx$

Let $\phi :[a,b] \rightarrow \Bbb R$ be a step function. Using $-|\phi| \leq \phi \leq |\phi|$ I need to conclude that $|\int^b_a \phi(x)dx| \leq \int^b_a|\phi(x)|dx$ I have no idea how to prove ...
53 views

### $\phi$ is a step function. Prove that $|\phi|$ is a step function

Let $\phi :[a,b] \rightarrow \Bbb R$ be a step function. I have to prove that $|\phi|$ is a step function. Here's how I prove it: Let $P$ be a partition $P=\{p_0,...,p_k\}$ on $[a,b]$, compatible ...
### Step functions: $\phi \leq \psi$ then $\int^b_a\phi(x)dx \leq \int^b_a\psi(x)dx$
Let $\phi,\psi :[a,b] \rightarrow \Bbb R$ be step functions. I have to show that if $\phi \leq \psi$ then $\int^b_a\phi(x)dx \leq \int^b_a\psi(x)dx$ How I go with proving this: Let $P$ be a ...