DERIVATIVE OF GREATEST INTEGER FUNCTION
For a greatest integer function [[ f(x) ]] defined in an interval [a,b];
⇒At the points which gives f(x)∈ Z, derivative does not exist.
⇒At the points which gives f(x)?Z, derivative is zero.
This situation is very readily on graphs:
The graphs of greatest integer function are always in this type.
Graph is constant in an interval with one or two discontinuities at the tips of the bars.
The constant regions are where f(x)z with a slope(derivative) zero.So derivative does not exist
The discontinuity points are where f(x)∈Z.Consider y=[]:
at x=+h function will be [[2+r]]with r>0 so y=2
at x=-h function will be [[2-r]] so y=1
Thus discontinuities occur and derivative vanishes.