What do we mean when we say that 3.14 is an approximation of *π* ? This means that on the real line these two numbers are ‘close’ to each other. The numbers −7.1 and −7 are closer to each other than the pair 10 and −1000. The modulus function is useful to measure the distance between two numbers. |*x*| is just the distance between origin and the point *x*. We formally define the modulus function || : ℝ → ℝ^{+} as follows.

|*x*| = *x*, *x* ≥ 0 (2.11)

= −*x*, *x* < 0. (2.12)

The distance between two points *x* and *y* is nothing but |*x* − *y*|. The statement |*x*| ≤ *a* is equivalent to the statement −*a* ≤ *x* ≤ *x*.

Show that the function *f* : ℝ → ℝ defined as *f*(*x*

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