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Modulus of Continuity of a Function Definition: Let f be defined on I. Set w_f(?) = w(f,?) = sup_{|x1-x2|<?} |f(x1) - f(x2)|. The function w_f(?) is called the modulus of continuity of f. Example: Let f(x) = x^2, I = (0,1), then w_f(?) = sup_{|x1-x2|<?} |x1^2 - x2^2|, where x1, x2 ? (0,1). |x1^2 - x2^2| = |x1 - x2| |x1 + x2| < ?(x1 + x2) ? ?(2 - 2?) = 2? - 2?^2, since 0 < x1 ? 1 - ?, 0 < x2 ? 1 ? x1 + x2 ? 2 - 2?. Thus w_f(?) = 2? - 2?^2; if ? = 1/n, 2/n - 1/n^2 ? 0. Exercises: Find the modulus of continuity of the following functions: 1) f(x) = 1/x, I = (0,1)

          Modulus of Continuity of a Function

Definition:
Let f be defined on I. Set w_f(?) = w(f,?) = sup_{|x1-x2|<?} |f(x1) - f(x2)|. The function w_f(?) is called the modulus of continuity of f.

Example: Let f(x) = x^2, I = (0,1), then w_f(?) = sup_{|x1-x2|<?} |x1^2 - x2^2|, where x1, x2 ? (0,1).

|x1^2 - x2^2| = |x1 - x2| |x1 + x2| < ?(x1 + x2) ? ?(2 - 2?) = 2? - 2?^2, since 0 < x1 ? 1 - ?, 0 < x2 ? 1 ? x1 + x2 ? 2 - 2?.

Thus w_f(?) = 2? - 2?^2; if ? = 1/n, 2/n - 1/n^2 ? 0.

Exercises: Find the modulus of continuity of the following functions:
1) f(x) = 1/x, I = (0,1)

Modulus of Continuity of a Function

Definition:
Let f be defined on I. Set wf(?) = w(f,?) = sup|x1-x2|<? |f(x1) - f(x2)|. The function wf(?) is called the modulus of continuity of f.

Example: Let f(x) = x^2, I = (0,1), then wf(?) = sup|x1-x2|<? |x1^2 - x2^2|, where x1, x2 ? (0,1).

|x1^2 - x2^2| = |x1 - x2| |x1 + x2| < ?(x1 + x2) ? ?(2 - 2?) = 2? - 2?^2, since 0 < x1 ? 1 - ?, 0 < x2 ? 1 ? x1 + x2 ? 2 - 2?.

Thus wf(?) = 2? - 2?^2; if ? = 1/n, 2/n - 1/n^2 ? 0.

Exercises: Find the modulus of continuity of the following functions:
1) f(x) = 1/x, I = (0,1)

Added by Charles H.

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