monotonically increasing

Monotonically increasing or strictly increasing?

Direct answer: Here, monotonically increasing means nondecreasing: for any x₁<x₂ in the interval, f(x₁)≤f(x₂). Equal outputs are allowed. A function is strictly increasing if f(x₁)<f(x₂) for every such pair. Terminology varies between textbooks, so check whether the definition in your question allows equality.

How to prove a function is increasing without derivatives

1. State the interval and check that the function is defined throughout it. An increasing interval is not necessarily the whole domain.
2. Choose any x₁<x₂ in that interval and calculate f(x₂)−f(x₁). The argument must work for every permitted pair, not just a few test values.
3. Factor or simplify the difference and use the interval conditions to determine its sign. A nonnegative difference proves nondecreasing behavior; a positive difference proves strict increase.
4. Give the interval with your conclusion. If the graph changes direction, discuss the intervals separately.

Worked example: f(x)=x²+2 on [0,∞)

Take any 0≤x₁<x₂. Then f(x₂)−f(x₁)=(x₂−x₁)(x₂+x₁). Both factors are positive: x₂−x₁>0, and x₂>x₁≥0 gives x₂+x₁>0. The difference is therefore positive. The table illustrates the result; the proof is the comparison of arbitrary inputs above.

The function is strictly increasing, and hence nondecreasing, on [0,∞). It is not increasing over all real numbers: −2<−1, but f(−2)=6>3=f(−1).

Common mistakes

• Assuming an increasing function must have positive values. The function f(x)=x−10 is strictly increasing on the real numbers even where its outputs are negative.
• Treating a flat section as strictly increasing. A constant function such as f(x)=4 is both nondecreasing and nonincreasing, but neither strictly increasing nor strictly decreasing.
• Using only the endpoints or a short table as a proof for a whole interval. The function might decrease between the sampled points.

Definition references and example notes

• Wolfram: nondecreasing, nonincreasing and strict monotonicity
• OpenStax Precalculus 2e: increasing and decreasing intervals

Last reviewed: 2026-09-10. The references explain the definitions. The worked example is an AICSCA practice example, not an official exam question.