A bounded derivative implies uniform continuity. However, there are certain instances of functions (most notably

) that are uniformly continuous and have unbounded derivatives. I attempt to strengthen the theorem below:

__Theorem__: An everywhere continuous function

is uniformly continuous on

if its derivative is bounded on

and

(

).

__Proof__: On

we have a continuous function defined on a compact set, so it is uniformly continuous and

... On

and

,

has a bounded derivative so there exist

and

such that... Therefore taking

proves that the entire function is uniformly continuous on

.