[Math] finding weird examples of continuous functions (or, maybe, they’re tricks?)

Question

I need to find examples of the following, but I feel like they're trick questions.

1. A unbounded function on a bounded closed interval (the function must be defined at every point in the interval)

I was thinking something like 1/x, but it's not defined at 0 (it'd need to be (0,1] ), so I don't see how I can provide something that is defined at the boundary points and unbounded on the interval. I would think I'd need something that blows up, but then it won't be defined at every point in the interval

2. f : [0, 1] → [0, 1], having the intermediate-value property but continuous at only one point

For a function to experience the intermediate-value property, doesn't it need to be continuous on its domain (the whole closed interval)? How can I provide an example when it's only continuous at one point?

Answer 1

f(x) = 0 if x = 0. f(x) = 1/x if x $ne$ 1.

Unbounded, defined everywhere, but not continuous.