I'm not sure how to approach these types of conceptual questions - would I have to use proofs? Suppose g(x) and h(x) are two continuous functions (on all real numbers) such that g(0) = h(0). Then if we define a function F by: F(x) = g(x) if x >= 0, h(x) if x <=0 (a) show that the function F is continuous for all real numbers. (b) If g′(0) and h′(0) exists, is it enough to say that F(x) is differentiable at x = 0? If not, give an example of such a case, i.e., give a function g(x) and another function h(x) such that g(0) = h(0), and g′(0) and h′(0) exist, but F′(0) does not. What condition will you then need for g′(0) and h′(0), such that F′(0) exists?  

Expert Answers

An illustration of the letter 'A' in a speech bubbles

(a) Clearly F is continuous on `(-oo,0),(0,oo)` since g and h are continuous. The only question is at 0.

Since h is continuous, `lim_(x->0^-)=h(0)`

Since g is continuous  `lim_(x->0^+)=g(0)`

But h(0)=F(0)=g(0). So F is continuous at 0 since the limit at 0 exists and equals the value of F(0).

(b) It is possible for g'(0) and h'(0) to exist, and for F to not be differentiable at 0. Consider h(x)=-x, g(x)=x. Then h'(0)=-1, g'(0)=1 and h(0)=0=g(0), and F is not differentiable since the derivative from the left is not equal to the derivative from the right.

In order to guarantee F differentiable, not only must h'(0),g'(0) exist, but h'(0) must equal g'(0).

Approved by eNotes Editorial Team

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial