Thread: Is this statement true or false?

Suppose that g is a continuous function on the interval [0,1] with g(0)=0. Define a sequence of functions on the interval [0,1] by . Does it follow that the sequence of functions converges uniformly on [0,1]? Either prove this is true or give an example of a function g for which the conclusion fails.

Any help on this?

Originally Posted by paupsers

Suppose that g is a continuous function on the interval [0,1] with g(0)=0. Define a sequence of functions on the interval [0,1] by . Does it follow that the sequence of functions converges uniformly on [0,1]? Either prove this is true or give an example of a function g for which the conclusion fails.

Any help on this?

The conclusion fails for , for example: because if were to converge uniformly, the limit function would have to be continuous, but it isn't continuous at .