Interval/set builder notation question

This is something I have seen a handful of math teachers do. I just want to distinguish if what they are doing is just unconventional.

The Issue:

Suppose you want to list the intervals of Increasing and decreasing of, say, x^2. Most books ( or people) would say:


Decreasing: ( -∞ ,0 ) or in set builder notation {x| -∞< x < 0 }
Increasing: (0,+∞) or in set builder notation {x | 0< x < +∞}

At x=0, x^2 is neither increasing or decreasing. Therefore, we use the parenthesis ( or < > ). But what I am seeing is the use of a bracket. Like so:

Decreasing: ( -∞ ,0] which translates to {x| -∞< x ≤ 0 }
Increasing: [0,+∞) which translates to {x| 0 ≤ x < +∞}

We know from calculus that the tangent slope at x=0 is 0 ( Neither increasing or decreasing). So the use of a bracket is not really necessary. However, I do not believe its wrong to use a bracket. Because a bracket translates to greater/less than "OR" equal. "OR" as in the number does not necessary have to satisfy both conditions.If we do this, then whats stopping us from saying " [-∞,0]" is the decreasing interval? I know x cannot actually equal - ∞.


Anyways, this is being done with all functions. Even with ones, say |x|, where the derivative does not exist at x=0. But x=0 is still included in the intervals.


Please elaborate on which is the preferred or right way of doing this.



∫ Σ → ∞ Π Δ Φ Ψ Ω Γ ∮ ∇∂ ± ∓ √ ∛ ∅ °

α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ σ τ υ φ χ ψ ω
⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁿ ₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋
⅓ ⅔ ⅕ ⅖ ⅗ ⅘ ⅙ ⅚ ⅛ ⅜ ⅝ ⅞
≡ ≈ ≅ ≠ ≤ ≥ · ~ ∤ ◅∈ ∉ ⊆ ⊂ ∪ ∩ ⊥ ô
≤ ≥ · ~ ∤ ◅∈ ∉ ⊆ ⊂ ∪ ∩ ⊥ ô

So the definition of an increasing function f on an interval, is "If x>y, then f(x) >f(y) in that interval."
(strictly increasing)
This means that whether or not you include the bracket is up to you. In your example of f(x)=x^2.