Sunday, 24 March 2013

real analysis - prove that a function f is continuous at x0 iff f(x0)=f(x+0)=f(x0)


prove that a function f is continuous at x0 iff limxx0f(x)=limxx+0f(x)=limxx0f(x)





By definition, a function f is said to be continuous at x=x0 if limxx0f(x)=f(x0)



It follows that for each ϵ>0 there is a δ>0 s.t. |f(x)f(x0)|<ϵ

whenever |xx0|<δ



Considering |xx0|<δ

is true
δ<xx0<δ

x0δ<x<x0+δ




Therefore,
On the right of x0, it follows that : x0x<x0+δ such that |f(x)f(x0)|<ϵ.
It shows that the function f is continuous from the right at x0,
Therefore:
limxx+0f(x)=limxx0f(x)



Similarly, on the left of x0, it follows that x0δ<xx0 such that |f(x)f(x0)|<ϵ.
It shows that the function f is continuous from the left at x0.
Therefore:
limxx0f(x)=limxx0f(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
limxx0f(x)=limxx+0f(x)=limxx0f(x)



Is this correct? anything could have been done differently? Any input is much appreciated

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