Is a Monotonic Function Always Increasing at x=0?

[player1_1_1]

Homework Statement

$$y=x^3$$

The Attempt at a Solution

I know that function is increasing when $$f'(x) > 0$$ but in $$x=0$$ there is $$f'(x) = 0$$, so is function increasing there or not? From definition I know that its increasing there, but how can I connect this with theorem that function is increasing when $$f'(x) > 0$$?

 

[vela]

What precisely is your definition of an increasing function?

 

[player1_1_1]

$$\forall_{x_1,x_2\in X} \left( x_1 < x_2 \Rightarrow f(x_1)<f(x_2) \right)$$

 

[losiu99]

If a function is non-decreasing (weakly increasing), and $$f(x_1)=f(x_2)$$, $$x_1\neq x_2$$, then f is constant on $$[x_1, x_2]$$, so a single zero of a derivative cannot spoil injectivity.

 

[player1_1_1]

aha, so if derivative is positive and 0 only for countable set of points the function will be increasing?

 

[losiu99]

Yes. It can probably be strenghtened a little bit, but that should be enough for all practical purpose.

 

[HallsofIvy]

$$\forall_{x_1,x_2\in X} \left( x_1 < x_2 \Rightarrow f(x_1)<f(x_2) \right)$$

With that definition, it makes no sense to talk about a function being "increasing" at a point. It is easy to prove that $y= x^3$ is increasing on any interval.

 

[player1_1_1]

thanks for answers. Is any definition which can define increasing in a point?

 

[player1_1_1]

stupid question, sorry, nvm