# Inflection point of non continuous or non differentiable function

## Homework Statement

three functions:
$$y=\begin{cases}\arctan \frac{1}{x}\ x\neq0\\ 0\ x=0\end{cases}$$
$$y=\frac{1}{x}, y=|x^2-1|$$ and what about inflection point?

## The Attempt at a Solution

first function is concave on left of 0, convex on right, so from definition it should be inflection point, but its not continuous in this point, a function need to be continuous in this place or not?
in 2, $$x=0$$ should be inflection point, but its not in the domain, so is there inflection point?
in 3, function is continuous in $$x=1$$ but not differentiable, is there inflection point or not?

up,.

LCKurtz
Homework Helper
Gold Member

## Homework Statement

three functions:
$$y=\begin{cases}\arctan \frac{1}{x}\ x\neq0\\ 0\ x=0\end{cases}$$
$$y=\frac{1}{x}, y=|x^2-1|$$ and what about inflection point?

## The Attempt at a Solution

first function is concave on left of 0, convex on right, so from definition it should be inflection point, but its not continuous in this point, a function need to be continuous in this place or not?
in 2, $$x=0$$ should be inflection point, but its not in the domain, so is there inflection point?
in 3, function is continuous in $$x=1$$ but not differentiable, is there inflection point or not?

It probably depends on the definition your text gives. Most say it must be a point on the graph where the concavity changes. That would rule out the first two. I would say the third qualifies because of the change in concavity at the point. But your mileage may vary.

thx!