In these exercises C0 C0a 6 M is the space of continuous re

In these exercises, C^0 = C^0([a, 6], M) is the space of continuous real-valued functions defined on the closed interval [a, 6]. It is equipped with the sup norm, ||f|| = sup {|f(x)|: X [a, b]}

Solution

> - at least 10 discontinuities

Are you ask, that if each fn has at least 10 discontinuities then so does f?

Similarily for finitely many discontinuities and countably many jump discontinuities

> - no jump discontinuities

This is a little different..

Suppose that each fn has no jump discontinuity. Can f have a jump discontinuity?

To this the answer depends on your definition of a \"jump discontinuity\".

For instance if you require both left and right limit to exist but be different from each other then the answer is

actually YES.

Ex) Consider fn to conferge uniformly to 0 on [0,1/2) and to 1 on [1/2,1]. But to also be discontinuous everywhere (use the rational trick).

i.e. On [0,1/2) set fn (x) = 1/n if x is a rational number and fn(x) = 0 if x is an irrational number.

and On [1/2,1] set fn (x) = 1 + 1/n if x is a rational number and f_n(x) = 1 if x is an irrational number.

Then each fn will have no jump discontinuities, since no limits (left or right) exist at any point.

However, its limit function is f(x) = 0 if x is in [0,1/2) and f(x) = 1 if x is in [1/2,1]. Which has a jump discontinuity at x = 1/2.