Discontinuous function example
Name: corey j catten
Date: Around 1995
What is a good, tangible, real-world example of a discontinuous function?
The amount of money in my checking account.
The acceleration of a typical automobile in heavy traffic.
When the domain of a function is naturally discrete then the function must
be discontinuous. Good examples are tables of values. In a lottery, the
probability of matching N numbers is a discrete function. When the domain
is continuous, such as time or mass, etc., then a naturally discrete range
(minimum gaps between units such as "cents" in asmith's account balance
example) implies a discontinuous function. Another standard example is
postage charges based on whole ounces even though weight is generally
considered a continuous domain. Here, I must disagree with jlu's accelera-
tion example. Jerky motion is not discontinuous in acceleration. For
example, striking a golf ball results in a great change in its acceleration
but still this occurs in some positive (small but not 0 ) time. Since we
yet await our first "jump to warp speed" experience, I do not think discon-
tinuous acceleration is practical in any common real model.
I am afraid that tee tries to intrude upon the idealizations of physics with
irrelevant real world considerations. The fact is that we could never solve
any problem if we could not idealize it. So when an ideal golf club strikes
an ideal golf ball, then the club head and the ball's surface are absolutely
rigid, and the acceleration of the golf ball is instantaneous. Thus,
acceleration may be a discontinuous function of the time. The contrast is
with position, which can never be discontinuous, otherwise a body could be
in two places at the same time.
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Update: June 2012