Examine the following functions for continuity.(a) `f(x)=x-5` (b) `f(x)=1/(x-5)` (c) `f(x)=(x^2-25)/(x+5)`(d) `f(x)=|x-5|`

Explanation

(a) f(x) = x - 5 is a polynomial function.
`:.` f(x) is contnuous for each real value .
(b) f(x) ` = (1)/(x - 5) , x ne 5 `
which is the quotient of two polynomials and is not
defined at x = 5
`:.` f(x) is contnuous for each real value of x (except x = 5).
(c) f(x) = `(x^(2) - 25)/(x + 5), x ne - 5`
`= ((x - 5) (x + 5))/(x + 5) = x -5`
which is a polynomial function .
`:.` f(x) is continous for all real value x = -5 .
(d) f(x) =`|x - 5| `
`{:={(x -5, if x ge 5),(5 - x, if x lt 5):}`
For x `ge ` 5, f (x) = x - 5
which is a polynomial function.
`:.` f(x) is continuous for x `ge `5.
For x `lt ` 5, f(x) = 5 - x
which is a polynomial function.
`:.` f(x) is ontinuous for x `lt` 5
Therefore, f(x) is continuous for all real value of x .