Question

Show that any linear function `f(x) = ax + b` is a continuous function in any `x_o in ℝ` ?

Answer 1

Use the definition of continuity.

Explanation:

Special case:
If a = 0, then `f` is constant throughout the real numbers.
In this case let `epsilon > 0`.
Then
`| f(x) - f(x_0) | = | b - b |`
`= 0`
This is less than `epsilon` regardless of how close `x` gets to `x_0`. Thus, every constant function is continuous.

Secondly, we observe that for `f(x)=ax + b`, and `a ne 0`,
we have `f(x_0) = ax_0 + b`.
Let `epsilon > 0`.
Then
`| f(x) - f(x_0) | = | ax + b - (ax_0 + b) |`
`= | ax + b - ax_0 - b | = | ax - ax_0 |`
`= | a ||x - x_0|`.

Choose `delta = epsilon/|a|`.

Then whenever `0 < | x - x_0 | < delta`,
we have
`| f(x) - f(x_0) | = | a ||x - x_0|`
`< | a |*epsilon/|a| = epsilon`.
Therefore `f` is continuous at `x_0`.