Problem:

For any two functions f and g in C([0,1],R), define

a. Prove that this defines a metric on C([0,1],R)

b. Prove that the following inequality relating this metric and the uniform metric:

c. Compare the concepts of convergence of a sequence of functions in this metric and in the uniform metric.

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Attempt:

a. First, C([0,1],R) denotes that the function is continuous.

We need to prove that

i. if and only if f = g. If f = g, then

ii. d*(f,g) = d*(g,f) (symmetry): This is true since , thus

iii. Triangle Inequality:

Thus, d*(f,g) is a metric on C([0,1],R)

b.We need to prove that

I'm unsure where to start. I try to divide both sides by (b-a) which yields

Here, I use the Mean Value Theorem for integrals, since the function on [a,b] is continuous, then there exists a c such that

c. I'm not sure.

Thank you for your time.