So basically, my metric space X is the set of all bounded functions from [0,1] to the reals and the metric is defined as follows: d(f,g)=sup|f(x)-g(x)| where x belongs to [0,1].(adsbygoogle = window.adsbygoogle || []).push({});

I want to prove that the set of all discontinuous bounded functions, D[0,1] in X is open.

My attempt - Start with an arbitrary function h in D[0,1]. h is discontinuous.

=> There exists a point y in [0,1] and r>0 such that for all s>0

|f(x) - f(y)| > r when |x-y|< s

Now, Consider the open ball B(h,r/3). This is the set of all functions f such that |f-h| < r/3 in the entire interval. I want to show that all the functions in this ball are discontinuous at the point y!

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# Prove C[0,1] is closed in B[0,1] (sup norm)

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