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Bounded Function Checker

Check if a function is bounded above or below on an interval

Select Function Type

sin(x) Quadratic ax^2+c Rational 1/x

f(x)=a*sin(x). Bounded for all x by [-|a|, |a|].

f(x)=sin(x)

Boundedness

Bounded above: exists U with f(x) <= U

Bounded below: exists L with f(x) >= L

Bounded: both above and below

Continuous on [a,b] always bounded

A bounded function has both a finite upper bound and a finite lower bound. Sine and cosine are bounded on R. Polynomials of even degree are bounded below (positive leading coeff) or above (negative). Rational functions may be bounded on closed intervals that avoid asymptotes.

⚠ Any continuous function on a closed interval is automatically bounded (Extreme Value Theorem). The real question is boundedness on the entire domain.

What Is Boundedness?

A function is bounded if its values stay within some finite range. Functions like sin(x) stay between -1 and 1 (bounded). Quadratic functions on all real numbers have either a lower bound (a>0) or an upper bound (a<0), but not both. Odd-degree polynomials are unbounded.

Bounded Functions

sin(x), cos(x): bounded by [-1,1]. arctan(x): bounded. e^(-x^2): bounded. Any function on a closed interval is bounded.

Unbounded Functions

x, x^3, e^x, ln(x), 1/x near 0: all unbounded on their natural domains. Polynomials of odd degree are unbounded.

One-Sided Bounds

x^2 for x>0 is bounded below (by 0) but not above. e^x is bounded below (by 0) but not above.

Supremum/Infimum

The least upper bound is the supremum. The greatest lower bound is the infimum. If attained, they are max/min.

Teaching Example: f(x)=2sin(x). Range = [-2,2]. Bounded above by 2, bounded below by -2. Bounded! f(x)=1/x on (0,inf): not bounded above (goes to inf), not bounded below (goes to -inf from left).

Applications

Analysis Integration Optimization Probability Physics

Frequently Asked Questions

What is bounded function?▼

Exists M with |f(x)| <= M. sin bounded by 1. x^2 not bounded above on R. Continuous on [a,b] always bounded.

Is sin bounded?▼

Yes, |sin(x)| <= 1 for all x. Both bounded above and below. Range = [-1,1].

Is x^2 bounded?▼

On R: bounded below by 0 but not bounded above (goes to inf). On [-5,5]: bounded by [0,25].

Closed interval bounded?▼

Any continuous function on [a,b] is bounded (Extreme Value Theorem). Always attains max and min.

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