Latest update

- Modular equation with exponents on the exponents
- What if metric space is replaced by arbitrary topological space, will the result still hold?
- How can I ansewer what statistical test to use?
- Should I augment validation set?
- Gradient for deep neural net in matrix notation?
- using regression model to optimize teams working on work items
- Using a zero slope coefficient predictor variable in multiple regression
- Gamma GLMM with log-link: how to report “rate of change” from exponentiated coefficients?
- Showing these statistics are ancillary
- Why old platformers are only scrolling right?
- Unity 3d. Upload mobile game to facebook playable ads
- Unity camera has no preview, and is not moving
- American pancakes: Why not substitute some wheat flour for starch?
- What is all default name of player in Just Dance game
- Patients are hungry and thirsty
- I need super power ideas for my game
- I have a problem with the Commands in Minecraft (execute/scoreboard)
- Record whole row in a changelog tab when a value in a specific column changes
- Inserting a row in Google Sheets does not copy the formula present in other rows
- How to log into Facebook

# How to make sense of this function in this proof of the Picard-Lindelöf theorem?

2018-07-22 02:48:14

See the pages below. I'm having trouble with the function $L_1$ defined at start of the second page. I have few questions:

On the 4th inequality in the second page, the author assumes that $L_1'(s) = L(s)$, but, without any other information, we can only do this if we know that $L$ is continuous (this is a condition in the Fundamental Theorem of Calculus). How do I show that $L$ is continuous?

If $L$ is continuous, isn't (2.26) automatically true?

This is sort of unrelated to the question, but $K: X \to X$ is defined by

$$

(Kx)(t) = x_0 + \int_{t_0}^t f(s, x(s))\ ds.

$$

Thanks in advance.