(a) Prove that the function F(x)=|x| is continuous on (−∞,∞)
(b) Prove that if f is a continuous function on an interval, then so is |f|.
(c) If |f| is continuous, does it follow that f is continuous as well? I so, prove it. If not, find an example.
(a) Based from the definition of absolute value,
F(x)=|x|={x if x≥0−x if x<0
According to the definition if the function is continuous everywhere, its left and right hand limits should be equal. So,
lim
It shows that \lim \limits_{x \to a^+} f(x) = \lim \limits_{x \to a^-} f(x), therefore F(x) is continuous on (-\infty, \infty)
(b) If f is continuous on an interval then |f| = f for f > 0 and |f| = -f for f < 0. Also, |f| = 0 if f = 0.
Therefore, if f is a continuous function, |f| is continuous as well.
(c) Suppose that f(x) = \left\{ \begin{array}{c} 1 & \text{ for } & x \geq 0 \\ -1 & \text{ for }& x < 0 \end{array} \right.
|f| is continuous everywhere such that if we plugin any positive or negative numbers inside the absolute value, we get
a positive value. However, f is not continuous at x = 0 because of jump discontinuity.
Thursday, April 20, 2017
Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 64
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