Question
give an example of a function that has:
a) only one point of discontinuity
b) exactly two ponts of discontinuity
c) an infinite number of discontinuity
give an example of a function that is:
a) continuous at every point
b) discontinous at every point
i am confused on how to give these examples
a) only one point of discontinuity
b) exactly two ponts of discontinuity
c) an infinite number of discontinuity
give an example of a function that is:
a) continuous at every point
b) discontinous at every point
i am confused on how to give these examples
Answers
Steve
think of a function which is undefined at one point.Usually that's a rational function, where it would evaluate to 0/0. So, try
y = (x^2-1)/(x-1).
y = x+1 for all real x, except where x=1. You can make it continuous by defining y(1)=2, which fills in the hole.
Extend this idea for two factors in the denominator
For c), try something like tan(x) which is discontinuities at all odd multiples of pi/2.
Every polynomial is continuous everywhere.
Something ghastly like
y=0 for rational x
y=1 for irrational x
is discontinuous everywhere. No matter how close you get to any rational number, there is an irrational number closer yet.
y = (x^2-1)/(x-1).
y = x+1 for all real x, except where x=1. You can make it continuous by defining y(1)=2, which fills in the hole.
Extend this idea for two factors in the denominator
For c), try something like tan(x) which is discontinuities at all odd multiples of pi/2.
Every polynomial is continuous everywhere.
Something ghastly like
y=0 for rational x
y=1 for irrational x
is discontinuous everywhere. No matter how close you get to any rational number, there is an irrational number closer yet.