[f(x)'^n continuous at a point (EXAM IN 4 hrs.)

by llursweetiell
Tags: continuity
 P: 1,105 [f(x)'^n continuous at a point (EXAM IN 4 hrs.) Sorry, the comment about the composition of functions threw me off. I should have known that the solvable interpretation is the right one. Anyways, you need to estimate |[f(x)]^n - [f(a)]^n| knowing that you can bind |f(x) - f(a)| by any positive number. But a^n - b^n has a very nice and symmetric formal factorization, namely $$a^n - b^n = (a-b)(a^{n-1} + a^{n-2}b + ... + b^{n-2}a + b^{n-1})$$ Thus you can replace a with f(x) and b with f(a) to get |[f(x)]^n - [f(a)]^n| equal to |f(x) - f(a)| times a bunch of terms within an absolute value (this is key) as a result of the factorization above. But now can you see what to bind |f(x) - f(a)| by to ensure that |[f(x)]^n - [f(a)]^n| is less than say, epsilon?
 P: 11 So i wrote, We know that f is continuous at a. to prove that g is continuous at a: let epsilon be greater than 0 and let delta > 0 such that d=epsilon/$$(f(x)^{n-1} + f(x)^{n-2}f(a) + ... + f(a)^{n-2}f(x) + f(a)^{n-1})$$ then, for |x-a| < delta, then |f(x)^n-f(a)^n|=$$f(x)^n - f(a)^n = (f(x)-f(a))(f(x)^{n-1} + f(x)^{n-2}f(a) + ... + f(a)^{n-2}f(x) + f(a)^{n-1})$$ < (f(x)-f(a)) * delta = epsilon. So g is continuous at a. Is that what you meant?
 P: 1,105 You have the right idea, but you don't actually get to choose delta here. Since f is continuous at a, there is already some delta for which |x-a| < delta implies $$|f(x) - f(a)| < \frac{\varepsilon}{(f(x)^{n-1} + f(x)^{n-2}f(a) + ... + f(a)^{n-2}f(x) + f(a)^{n-1})}.$$ But this is exactly what we need.
 P: 11 [QUOTE= $$|f(x) - f(a)| < \frac{\varepsilon}{(f(x)^{n-1} + f(x)^{n-2}f(a) + ... + f(a)^{n-2}f(x) + f(a)^{n-1})}.$$ QUOTE] do you mean f(x)^n-f(a)^n < all of the factorization?
 P: 1,105 Eh, perhaps it would be clearer if I just pieced together the proof. Let epsilon > 0 be given. By the continuity of f at a, there exists some delta > 0 such that |x-a| < delta implies $$|f(x) - f(a)| < \frac{\varepsilon}{|f(x)^{n-1} + f(x)^{n-2}f(a) + ... + f(a)^{n-2}f(x) + f(a)^{n-1}|}.$$ But then $$|f(x)^n - f(a)^n| = |f(x)-f(a)||f(x)^{n-1} + f(x)^{n-2}f(a) + ... + f(a)^{n-2}f(x) + f(a)^{n-1}| < \varepsilon.$$