1

I hope that you could help me with these two questions:

  1. Prove that if $\ f(x)$ is continuous at $x=a$, and $\ f(a) \neq 0$, then $1/\ f(x)$ is continuous at $x=a$.

  2. Prove that if $\ f(x)$ and $g(x)$ are continuous at $x=a$, and $g(a) \neq 0$, then $\ f(x)/g(x)$ is continuous at $x=a$.

alexwlchan
  • 2,060
Jason Lim
  • 21
  • 5
    Posting a question in the language appropriate to assigning homework is sometimes frowned on here. Adding your own thoughts on this or specifying at what point you ran into difficulty in trying to solve this problem are usually appropriate for posting here. – Michael Hardy Aug 13 '14 at 21:10

1 Answers1

1

Hint: First prove that if $g$ is continuous at $x_0$ and $h$ is continuous at $g(x_0)$ then $h\circ g$ is continuous at $x_0$. Then use the fact that $u(x)=\frac{1}{x} $ is continuous on $\mathbb R \setminus \{0\}$.

Here's a start:

Suppose $g$ is continuous at $x_0$ and $h$ is continuous at $g(x_0)$.Let $\epsilon > 0$, then since $h$ is continuous at $g(x_0)$, there is some $\tilde \delta> 0$ such that $$|y-g(x_0)|< \tilde \delta \implies |h(y)-h(g(x_0))|< \epsilon.$$ Now, $g$ is continuous at $x_0$ and so there is some $\delta > 0$ such that $$ |x-x_0|< \delta \implies |g(x)-g(x_0)|< \tilde \delta,$$ it follows that $$|x-x_0|< \delta \implies |h(g(x))-h(g(x_0))|< \epsilon,$$ i.e. $h\circ g$ is continuous at $x_0$.

Now, can you apply these results to concludes?

Community
  • 1
Surb
  • 55,662