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

Answer 1

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.