Question

Find a monotonic function on [0, 1] with discontinuities at 1/3,
2/3, and 3/4 only

Answer 1

To find a monotonic function on [0, 1] with discontinuities at specific points, you can define different functions on each sub-interval.

Step-by-step explanation:

To find a monotonic function on [0, 1] with discontinuities at 1/3, 2/3, and 3/4 only, we can break up the interval [0, 1] into sub-intervals and define different functions on each sub-interval.

For example:
On the interval [0, 1/3), we can define a constant function f(x) = 0.
On the interval (1/3, 2/3), we can define another constant function g(x) = 1.
On the interval (2/3, 3/4), we can define h(x) = x^2.
On the interval (3/4, 1], we can define a third constant function k(x) = 2.

By doing this, we have defined a function with discontinuities at 1/3, 2/3, and 3/4 only, and the function is monotonic because it either stays constant or increases within each sub-interval.