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# Analysis Continuity Watch

1. I've come across the following question and don't have any idea where to start with it.. so any help or pointers would be greatly appreciated!

Let deonte the space of continuously differentiable functions on the interval [0,1] with the supremum norm inherited from C([0,1]).

Which of the following maps are continuous? Give a proof or counterexample:

1) given by

2) given by

3 given by

Please help me to try and understand what I need to do
2. (Original post by gemma331)
I've come across the following question and don't have any idea where to start with it.. so any help or pointers would be greatly appreciated!

Let deonte the space of continuously differentiable functions on the interval [0,1] with the supremum norm inherited from C([0,1]).

Which of the following maps are continuous? Give a proof or counterexample:

1) given by

2) given by

3 given by

Please help me to try and understand what I need to do
There are a couple of ways to go about this, depending on what kind of course these questions are from. First, do you know what it means for a function to be continuous? It's possible to show these maps are (not) continuous straight from the basic definition of continuity.

If you've come across the idea of normed vector spaces before, then you might know that a linear map between two normed vector spaces is continuous if and only if it is bounded* - this is another way you can approach the question, and is a lot less work than the first approach.

*for a map f:X -> Y to be bounded in this sense, it means that there exists a constant K>0 such that for any vector x in X with norm less than or equal to 1, the norm of f(x) is less than or equal to K.

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Updated: March 27, 2013
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