show that if f is a function and x fx is a point on f and x

show that if f is a function and ( x, f(x)) is a point on f and x is not a limit point of the domain of f, then f is continous at ( x, f(x))

Solution

f is a function

f(x) is well defined

Since x is not a limit point of domain of f, the neighbourhood containing x points also have f(x)

In other words limit x tends x+ f and x- f both exist and equal to f(x)

Hence the limit exists for x tends to x and equal to f(x)

So f has to be continuous at x.