# Prove that the function f(x) = ax +b is uniformly continuous in R [duplicate]

Clearly, this problem requires applying epsilon, delta definition of uniform continuity. im having trouble finding values that will work here and hence, writing the proof in general.

• See the answers to this question. – Dietrich Burde Dec 10 '16 at 19:23
• What have you tried so far? What criteria must be satisfied to have uniform continuity? You're a lot more likely to get help if you've shown that you've at least made an effort. – NNN Dec 10 '16 at 19:50

We say $f$ is uniformly continuous if

$\forall \, \varepsilon > 0 \; \exists \, \delta > 0 \,$ such that $\forall x,y\in \mathbb{R}$ we have $|x-y|<\delta \Rightarrow |f(x) - f(y)| < \varepsilon$.

If $a=0$, then $|f(x)-f(y)| = |b-b| = 0 < \varepsilon$ for any $\varepsilon$, so any $\delta$ works.

Now let $\varepsilon>0$ and suppose $a \neq 0$. Let $\delta = \frac{\varepsilon}{|a|}$ and suppose $|x-y|<\delta$. Then

$$|f(x) - f(y)| = |ax + b - (ay + b)| = |ax - ay| = |a||x-y| < |a|\delta = |a|\frac{\varepsilon}{|a|} = \varepsilon$$

and we're done.

I figured out two values that work but not sure if this proof makes sense here

• Only problems: $a$ might be negative, in which case you want to divide by the absolute value, and $a$ might be zero, in which case you want a slightly different approach. – Ian Dec 10 '16 at 19:52
• Should I use the absolute value of (a) here then? instead of just a? – Aggrawal Puja Dec 10 '16 at 19:53
• That'll handle the nonzero case yes. – Ian Dec 10 '16 at 20:58