Question

Prove that greatest integer function is continuous at all the points except at integer points.

Answer 1

Checking continuity when x=2

(i) f(x)=x for all x $\epsilon$ R

By 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$\to$2-h):f(x)=[2-h]=1 -------(2)

[Since (2-h) lies between 1 and 2 and the least being 1]

right side limit

(x$\to$2+h):f(x)=[2+h]=1 -------(3)

[Since (2+h) lies between 2 and 3 and the least being 2]

(iii) Thus from above 3 equations left side limit is not equal to right side limit.

So, limit of function does not exist.

Hence, it is discontinuous at x=2.

So, greatest integer function is not constant at all points.