2012-10-08T08:24:31-04:00
Suppose f is a function such that
f(x) = x^2 x element Q,
0 x element R − Q.
Is f continuous at c = 0? Is f differentiable at c = 0?
1
2012-10-09T08:47:21-0400
1) The function is continuous at c=0. Indeed, fix any epsilon>0 and put delta = epsilon. We can assume that epsilon<1. Then for any x with |x|<delta we have that |f(x)| < epsilon. Indeed, if x is an element of Q, then |f(x)| = |x^2| < |x| < epsilon, since |x|<1. if x is an element of R-Q, then |f(x)| = 0 < epsilon. Thus f is continuous at c=0. 2) f is also differentiable at c=0, in the sense that lim_{x->0} ( f(x)-f(0) ) / x = 0. Indeed, fix epsilon>0 and put delta = epsilon. Then for any x with |x|<delta we have that | (f(x)-f(0)) / x | < epsilon. Indeed, if x belongs to Q-0, then | [ f(x)-f(0) ] / x | = | (x^2 - 0) /x | = |x| < epsilon and if x belongs to R-Q, then f(x)=0, and | (f(x)-f(0)) / x | = | (0 - 0)/x | = 0 < epsilon Thus lim_{x->0} [ f(x)-f(0) ] / x = 0. 3) On the other hand f is not continuous and not differentiable at any other point c<>0. Indeed, if c<>0, then take two sequences (a_i) and (b_i) converging to c such that each a_i is raitonal, i.e. belogns to Q and each b_i is irrational, i.e. belogns to R-Q. Then lim_{i -> infinity} f(a_i) = lim_{i -> infinity} a_i^2 = c^2, while lim_{i -> infinity} f(b_i) = lim_{i -> infinity} 0 = 0 <> c^2. So f is not continuous at c, and therefore it is not differentaible at c as well.
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