Question

is it possible to have a function f defined on [ 5 , 6 ] and meets the given conditions? f is continuous on [ 5 , 6 ], takes on the values − 5 and 5 but does not take on the value 0.

Answer 1

No.

The intermediate value theorem (IVT) says that if
`f(x)` is continuous on
`[a,b]`, and if
`d` is a number between

`\min\{f(a),f(b)\} \le d \le \max\{f(a),f(b)\}`

then there is some
`c\in(a,b)` such that
`f(c) = d`.

In this case we have for all
`x\in[5,6]`,

`-5 \le f(x) \le 5`

0 falls in this range, so by continuity of
`f` and the IVT, there must be some number
`x\in(5,6)` such that
`f(x) = 0`.