Lebl-Basic-Analysis-I · Problem 6.3.5

Sequences of Functions

Mathematics · lipschitz-in-second-variable-continuity

Problem

Suppose

F : I \times J \to \mathbb{R}

is a function that is continuous in the first variable, that is, for every fixed

y

the function that takes

x

F(x,y)

is continuous. Further, suppose

F

is Lipschitz in the second variable, that is, there exists a number

L

such that

\left|F(x,y) - F(x,z)\right| \leq L\left|y-z\right| \quad \text{for all } y,z \in J, x \in I.

Show that

F

is continuous as a function of two variables. Therefore, the hypotheses in the theorem could be made even weaker.