- #1

Felafel

- 171

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Hi! I think I've managed to solve this problem, but I'd like it to be checked

show that if $$f : A\subset \mathbb{R}\to \mathbb{R}$$ and has both right derivative:

$$f_{+}'(x_0),$$

and left derivative

$$f_{-}'(x_0)$$

in $$x_0\in A$$, then $$f$$

is continuos in

$$x_0.$$

Let's assume $$f_{+}' > f_{-}'$$, as the derivative exists, it means it is $$< \infty$$.

Therefore, $$|f(x)-f(y)|≤f_{+}'|(x-y)|$$ is a Lipschitz equation, and for

$$|f(x)-f(x_0)|≤f_{+}'|(x-x_0)|$$

it is a lipschitz equation in x_0.

Thus, for the lipschitz equation properties, the function is continuos in $$x_0$$

## Homework Statement

show that if $$f : A\subset \mathbb{R}\to \mathbb{R}$$ and has both right derivative:

$$f_{+}'(x_0),$$

and left derivative

$$f_{-}'(x_0)$$

in $$x_0\in A$$, then $$f$$

is continuos in

$$x_0.$$

## The Attempt at a Solution

Let's assume $$f_{+}' > f_{-}'$$, as the derivative exists, it means it is $$< \infty$$.

Therefore, $$|f(x)-f(y)|≤f_{+}'|(x-y)|$$ is a Lipschitz equation, and for

$$|f(x)-f(x_0)|≤f_{+}'|(x-x_0)|$$

it is a lipschitz equation in x_0.

Thus, for the lipschitz equation properties, the function is continuos in $$x_0$$

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