Prove that the greatest integer function $\lfloor x\rfloor$ is continuous at all points except at integer points.

Question

Prove that the greatest integer function $\lfloor x\rfloor$ is continuous at all points except at integer points.

I was solving this function , now the question that arises is that I was solving this using an example i.e. A numerical value, but my teacher keeps saying that it's wrong or I have to solve it using constants such as k... Etc. Is this method wrong according to u?

i) f(x) = [x], for all x in R ==> By the definition of greatest integer function: If x lies between two successive integers, then f(x) = least integer of them.

ii) So, at x = 2, f(x) = [2] = 2 -------- (1)

Left side limit (x ---> 2-h): f(x) = [2 - h] = 1 ----- (2) {Since (2 - h) lies between 1 & 2; and the least being 1}

Right side limit (x --> 2+h): f(x) = [2 + h] = 2 -------- (3) {Since (2+h) lies between 2 & 3; and the least being 2}

iii) Thus from the above 3 equations, left side limit is not equal to right side limit. So limit of the function does not exist. Hence it is discontinuous at x = 2 So this is not derivable at x = 2 Hence Proved.

Answer 1

Your argument is correct and valid. However, you still have a way to go, since...

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You proved:

The function is discontinuous at $2$.

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You have to prove:

1. The function is discontinuous at all integer points.
2. The function is continous at points that are not integers