Question

Examine the continuity of the function:

f: [ 1,3]→R defined by :

f(x)= [x]/3x-1 where [x] denotes the greatest integer function

Answer 1

$f(x)=\left\{\begin{array}{cc} \frac{1}{3x-1}&,x=1\\ \frac{2}{3x-1}&,1<x\leq2\\ \frac{3}{3x-1}&,2<x\leq3\\ \end{array} \right.$

function is continuous on (1,2) and on (2,3), for point 1 the finction has a side-altar on the right equal to 1 and the value is 0.5, for point 2 there is no side-altar, the side-altar on the left is 0.4 and the side-altar on the right is 0.6.

for point 3, the aisle on the left coincides with the value.

The function has break points.