Question

13.4
a) Let \(f\) be continuous in \(x_{0} \in \mathbb{R}\) with \(f(x_{0}) = 0\), and \(g\) bounded. Show, that the product \(fg\) also
is continuous in \(x_{0}\).

Answer 1

\[
| f g(x)- fg(x0)|=| f(x) g(x)- f(x0)f (x0)||=|f(x)||g(x)|\le M |f(x)|\\
\lim_{x\to x0} | f g(x)- fg(x0)| \le M \lim_{x\to x0} |f(x)| M |f(x0)|=0
\]

Answer 2

M is the bound og |g|

Answer 3

of

Answer 4

thank you very much mr Elias

Answer 5

in the first line f(x0) (f(x0) should be f(x0)g(x0)

Answer 6

ok no problem thank you :)

Answer 7

yw