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

Hint: We will be using the concepts of continuity for the question given to us, also we will be using the concept of functions.We know that a greatest integer function by definition is if $x$ lies between two successive integers then $f\left( x \right)=least$ .So, \[f\left( x \right)=\left[ x \right]\]

Complete step by step answer:
Now we have to prove that \[f\left( x \right)\] is discontinuous at all integer points. For this we take an integer \[x\in \] .
Now we know,
$f(x)=x.........(i)$
Now, we will be using a test of continuity by checking if the left hand side limit is equal to the right hand side or not.
So, In left hand side limit we have
$\underset{x\to x-h}{\mathop{\lim }}\,f\left( x \right)=\left[ x-h \right]=x-1.........(ii)$
So, in right hand side limit we have,
$\underset{x\to x+h}{\mathop{\lim }}\,f\left( x \right)=\left[ x+h \right]=x+1.......(iii)$
Now we have from (i), (ii) and (iii) that,
$L.H.L\ne R.H.L\ne f\left( x \right)$
Since L.H.L, R.H.L and the value of function at any integer $n\in $ are not equal therefore the greatest integer function is not continuous at integer points.

Note: To solve these types of questions one must have a clear understanding of the limit of a function and its rule. Also, one must have a basic understanding of functions.One must know how to operate left hand limit and right hand limit.