1. The problem statement, all variables and given/known data
Let [itex] [a,b] \subset \mathbb{R} [/itex] be a compact interval and t0 [itex]\in [a,b] [/itex] fixed. Show that the set [itex] S = {f \in C[a,b] | f(t_0) = 0} [/itex] is not dense in the space [itex]C[a,b][/itex] (with the sup-norm).
2. Relevant equations
Dense set: http://ift.tt/XgkiVj
sup - norm: http://ift.tt/1mQGBfO
3. The attempt at a solution
I tried to take function f from S and function g from C[a,b] and calculate the sup-norm of the difference of the functions and make it bigger than some number. However I am not able to do so.. I'm not even sure if my approach is correct here. What should be my strategy?
Let [itex] [a,b] \subset \mathbb{R} [/itex] be a compact interval and t0 [itex]\in [a,b] [/itex] fixed. Show that the set [itex] S = {f \in C[a,b] | f(t_0) = 0} [/itex] is not dense in the space [itex]C[a,b][/itex] (with the sup-norm).
2. Relevant equations
Dense set: http://ift.tt/XgkiVj
sup - norm: http://ift.tt/1mQGBfO
3. The attempt at a solution
I tried to take function f from S and function g from C[a,b] and calculate the sup-norm of the difference of the functions and make it bigger than some number. However I am not able to do so.. I'm not even sure if my approach is correct here. What should be my strategy?
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