If exists then is it proper to say that

provided f(x) is continuous at x=a.

I read this somewhere and do not believe it to be authentic.

I have come across many examples which appear to support the above statement but then there are other examples which do not support this.

If it is true then is there a rigorous proof in support of this.

One example that I have is as follows:

In this case is

Another example that i have is:

In this case is

Would be really grateful if someone verifies this for me

Thanx in advance

Also i must mention that [x] denotes the greatest integer less than or equal to x