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

1 Answer
Feb 11, 2018

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.