Suppose that is continuous, where is a region in . Let be a simple closed curve in . Show that for each , is continuous on .
Let . Suppose that there is such that . I need to show .
.
I got stuck here... can I get some help please?
Suppose that is continuous, where is a region in . Let be a simple closed curve in . Show that for each , is continuous on .
Let . Suppose that there is such that . I need to show .
.
I got stuck here... can I get some help please?
There's also a more general proof using uniform continuity; but let's do it your way.
Remember . We have
hence, using the above formula,
.
Since the curve is a bounded closed set (thus a compact set), you can choose such that the closed ball does not meet . Then we can find such that, for any in this ball, for all . Using these bounds in the previous inequality enables to bound the right-hand side by for some constant , and to conclude quickly from there.