Question

prove that the greater integer function is discontinuous ​

Answer 1

Step-by-step explanation:

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

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

Left side limit ( x----> 2h): 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}

||. 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.