# Math Help - Continuity of functions - Real Analysis

1. ## Continuity of functions - Real Analysis

I have a homework assignement due tomorrow morning and I have some idea on how to do this problem but I am not very sure.

Define functions f and g on [-1,1] by

$
f(x) = xcos(1/x), if \quad x \neq 0;$

$
$

$
g(x) = cos(1/x), if \quad x \neq 0;$

$
$

Prove that f is continous at 0 and that g is not continuous at 0. Explain why these functions are continuous at every other point in [-1,1].

So I was thinking for f(x) just using the definition of continuity:

$\left| f(0) - f(x) \right| = \left| 0 - xcos(1/x) \right| = \left| xcos(1/x)\right| < \left| x \right| \left| cos(1/x) \right| < \left| x \right| < \epsilon$
Choose $\delta = \epsilon$
$
\left| x- 0 \right| < \delta
$

Ok this proves that f is continuous at 0 right??

I am not entirely sure how to prove g(x) isn't continuous but also that these functions are continuous at every other points!
Any suggestion??

2. Consider the sequence $x_n = \frac{1}{{2\pi n}} \Rightarrow \left( {x_n } \right) \to 0$.
What is the value of $\left( {\forall n} \right)\left[ {g(x_n ) = ?} \right]$

3. Ok, so I was able to prove it that g is not continuous at 0 but now I do I prove that f and g are continuous for every x in the interval???

4. Originally Posted by ynn6871
that g is not continuous at 0 but now I do I prove that f and g are continuous for every x in the interval???
Originally Posted by ynn6871
I prove that f and g are continuous for every x in the interval???
Well you don’t do that because it is not true of $g$.
You proved that $g$ is not continuous at $x=0$ so it cannot be continuous for all $x$.

You also proved that $f$ is continuous at $x=0$.
Note that if $x \ne 0$ then both functions $x\,\& \,\cos \left( {1/x} \right)$ are continuous.
The product of two continuous functions is continuous.