Tuesday, 21 January 2014

real analysis - f,g continuous on closed finite interval, f(x)>g(x) for all x from the interval, implies f(x)>g(x)+alpha for some alpha>0?



The question:



Let f(x) and g(x) be continuous on the closed, finite interval [a,b] with f(x)>g(x) for all x[a,b]. Prove that there exists an α>0 such that f(x)>α+g(x) for all x[a,b].



I tried using ϵδ continuity as well as sequential continuity but didn't get anywhere. Any suggestions would be very helpful!



(Note: these are real-valued functions only.)



Answer



Hint: Does α=min satisfy f(x) > g(x) + \alpha for x \in [a,b]?


No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...