Is the function continuous at x=2 in f(x)= (x^2-4)/(x-2)?

[klmdad]

My Question is what 3 things are necessary for a function to be continuous at c?
I have Right hand must exist, left hand must exist
and right and left hands must be equal. I was told that this is worng.

 

[dagger32]

f(c) must exist

limit of f(x) as x approaches c must exist

limit of f(x) as x approaches c must equal f(c)


you were confusing continuity with the properties of one-sided limits that states if the right hand limit equals the left hand limit, then the limit as a whole exsists.

 

[ToxicBug]

A function that is differentiable is always continuous
A function that is continouos is not always differentiable

Therefore, your left and right hands notion is wrong. Think about the corner. It is cts but not diffble.

 

[HallsofIvy]

If f(x)= x for x< 0
and f(x)= x for x> 0

Then both left and right sided limits exist and are both 0, of course, but f is not continuous at x= 0 because f(0) is not defined at x= 0.

That's called a "removable discontinuity" because you can make the function continuous just by defining f(0) to be 0, but then you have a different function.

Exercise: Is f(x)= (x²- 4)/(x- 2) continuous at x= 2?