how do i find where all the values of x where f is discontinuous?

Question

where are all the values of x where f is discontinuous? also, which discontinuities are removable or unremovable?

f(x)=
x^2, x<2
3, x=2
3x-2, 2 2x, x>= 4

Answer 1

f is a piecewise continuous function, as it is made up of pieces each of which is continuous. So the only places f can possibly be discontinuous is at the boundary points x = 2 and 4. So those are the only points you need to check.

The rule for continuity at a point x = a is that lim f must exist at x = a, f must be defined at a, and its value f(a) must actually be equal to the limit. You need all three of these things, and only these. If the limit exists but the function value does not equal it, then you have a removable discontinuity. Otherwise the discontinuity is not removable.

Near x = 2, f approaches 4 as you approach from the left and also approaches 4 from the right, but f(2) = 3. So 2 is a removable discontinuity. You can remove the discontinuity merely by redefining f(2) = 4.

Near x = 4, f approaches 10 as you come in from the left, and approaches 8 as you come from the right. So the function has a finite jump at x = 4, which is a nonremovable discontinuity.

Answer 2

All the elementary functions are continuous.

The points you need to check are those where the elementary function changes, those points are 2 and 4

Let us check the point x=2:

lim (x -> 2, x<2)f(x) = lim (x -> 2)x^2 = 2^2 = 4

lim (x->2, x>2)f(x) = lim (x->2, x>2) 3x - 2 = 3x - 2 = 4

But, f(2) = 3, thus at 2 f is discontinuous.
However, if you define f(2) to be 4, then f will be continue at x=2.
Thus, x=2 is a removable discontinuity point.

Let's check f(4)

f(4) = 2*4 = 8

lim (x->4, x<4) f(x) = lim (x->4, x<4) 3x - 2 = 3*4 - 2 = 10

Thus, x=4 is unremovable.

Answer 3

Put in the boundary values for x at each place where the function changes, for the old one and the new one, and see if it's the same (continuous) or different (discontinuous). Here for example, it changes at 2

x^2 would be 4 but at 2 it becomes 3 so it's discontinuous at x=2
It changes next at 4. 3x-2 would be 10 and 2x would be 8 so again it's discontinuous