A function's continuity at a specific point hinges on three key conditions. First, the function's limit must exist as it approaches the point from both the left and the right. Second, the value of the function at that point must be defined.
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.
