prove that a function f is continuous at x0 iff limx→x−0f(x)=limx→x+0f(x)=limx→x0f(x)
By definition, a function f is said to be continuous at x=x0 if limx→x0f(x)=f(x0)
It follows that for each ϵ>0 there is a δ>0 s.t. |f(x)−f(x0)|<ϵ
Considering |x−x0|<δ
−δ<x−x0<δ
x0−δ<x<x0+δ
Therefore,
On the right of x0, it follows that : x0≤x<x0+δ such that |f(x)−f(x0)|<ϵ.
It shows that the function f is continuous from the right at x0,
Therefore:
limx→x+0f(x)=limx→x0f(x)
Similarly, on the left of x0, it follows that x0−δ<x≤x0 such that |f(x)−f(x0)|<ϵ.
It shows that the function f is continuous from the left at x0.
Therefore:
limx→x−0f(x)=limx→x0f(x)
We can conclude then that if f is continuous at x0, f has to be continuous from both the left and the right at x0,It follows that
limx→x−0f(x)=limx→x+0f(x)=limx→x0f(x)
Is this correct? anything could have been done differently? Any input is much appreciated
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