3 part definition of Continuity

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3 part definition of Continuity

The limit of f(x) as x approaches a=f(a)

If they are equal and this theorem can be applied, summarize your work in a short explanation.

If you get that f(a) isn't defined or the limit does not exist (DNE) then this function isn't continuous at that x-value.

1) Although you might get two different numbers for f(a) and the limit, the numbers HAVE TO BE EQUAL

f(a) is defined

If f(x) has a hole and/or a vertical asymptote, f(x) is not continuous at x=a

f(x) has no asymptotes at x=a

f(x) has no holes at x=a

What's the importance of this?

If one of these main ideas are not met, we know that there is some inconsistencies in the graph (discontinuities).

This determines a whole world of things like for instance, if we can find the derivative at that certain x-value (differentiability implies continuity).

If all three of the main ideas are met, f(x) is continuous at this x-value (x=a).

The limit of f(x) as x approaches a exists

3) The limits from the left and right equal each other.

2) The limit of f(x) as x approaches a FROM THE RIGHT exists.

1) The limit of f(x) as x approaches a FROM THE LEFT exists.