Question

How do you explain why `f(x) = x^2 +1` is continuous at x = 2?

Answer 1

There is no 'hole' in the graph and you can find the derivative at x =2.

Explanation:

A graph is considered continuous between any range `[a, b]` if you can draw it without lifting your pencil off the graph, meaning there are no holes - or locations where the value is undefined.

We can also try and take the derivative which would be `2x` then plug in `2` for `x` and get a slope of `4`. Which would mean at `x=2` there is a slope `4`, meaning the function exists at `2`.