Question

Show that the function defined by g(x)=x-[x] is discontinuous at all integral points.
 Here [x] denotes the greatest integer less than or equal to x.

Answer 1

Hint-To solve these types of problems calculate the value of LHL and RHL and show
 that the value of $LHL\ne RHL$which means to say that they are discontinuous.

The given function is g(x)=x-[x]
In this function let us consider an integer n and solve it
On substituting the value of n in the equation, we get
 g(n)=n-[n]=n-n=0
Now let us take the LHL and RHL of this equation,
We get LHL at x=n=$\underset{x\to n^{-}}{lim}g(x)=\underset{x\to n^{-}}{lim}(x-[x])=n-(n-1)=1$
             RHL at x=n=$\underset{x\to n^{+}}{lim}g(x)=\underset{x\to n^{+}}{lim}(x-[x])=n-n=0$
So, from this we can clearly observe that the value of $LHL\ne RHL$
If, for a function $LHL\ne RHL$, then we can say that the function is discontinuous
So, we can say that g(x)=x-[x] is discontinuous at all integral points
Note: If a similar type of question is asked to show that the functions are continuous then
 show that LHL=RHL , which means to say that the function is continuous.