Describe pictorially on a graph the set of functions g in Ca

Describe pictorially (on a graph) the set of functions g in C[a,b] whose distance from a given function f is less than or equal to 1 for each of the following metrics :

(3) the discrete metric

Describe pictorially (on a graph) the set of functions g in C[a,b] whose distance from a given function f is less than or equal to 1 for each of the following metrics : (1) the integral metric Z(f,g)= sup{|f(x)-g(x)|: x E [a,b]} (3) the discrete metric int a between b |f(x)-g(x)|dx (2) the supremum metric Z\'=

Solution

1) integral metric zf(g) <=

Thus the integral <= (b-a)

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2) The supremum is = integral 1 dx = (b-a)

3) The discrete metic = |F(b)-F(a)_G(b)+G(a)|

Where F is integrla of f and G is integral of G