Thread: Prove that h is not continuous at 0

1. Prove that h is not continuous at 0

Let L subset to R and define

h(x) = {sin(1/x), x not = 0
h(x) = { L, x = 0

Prove that h is not continuous at 0. (Hint: Prove by contradiction,
taking epsilon = 1/2 and use question 1)

question 1:

Suppose delta > 0.

(a) Prove that there exists a positive integer n such that

0 < 1/((4n+1)pi/2) < delta

and sin ((4n+1)pi/2) = 1

(b) Prove that there exists a positive integer m such that

0 < 1/((4m+3)pi/2) < delta

and sin ((4m+3)pi/2) = -1

Can anyone help me with this, I appreaciate it

2. Originally Posted by 450081592
Let L subset to R and define

h(x) = {sin(1/x), x not = 0
h(x) = { L, x = 0

Prove that h is not continuous at 0. (Hint: Prove by contradiction,
taking epsilon = 1/2 and use question 1)

question 1:

Suppose delta > 0.

(a) Prove that there exists a positive integer n such that

0 < 1/((4n+1)pi/2) < delta

and sin ((4n+1)pi/2) = 1

(b) Prove that there exists a positive integer m such that

0 < 1/((4m+3)pi/2) < delta

and sin ((4m+3)pi/2) = -1

Can anyone help me with this, I appreaciate it
Remember that to prove $\displaystyle h$ is not continous at $\displaystyle 0$ we must show that there exist an $\displaystyle \epsilon >0$ such that for all $\displaystyle \delta >0$ there exists an $\displaystyle x \in \mathbb{R}$ such that $\displaystyle \vert x \vert < \delta$ and $\displaystyle \vert L - \sin (1/x) \vert \geq \epsilon$. As you're advised take $\displaystyle \epsilon=1/2$ and assume $\displaystyle L \in [0, \infty)$ then by (b) of question 1 we have an $\displaystyle x$ such that $\displaystyle 0<x<\delta$ and $\displaystyle \sin (1/x)=-1$, now since $\displaystyle L\geq 0$ we have $\displaystyle \vert L-\sin (1/x) \vert >1>1/2$. If $\displaystyle L \in (-\infty,0)$ use (a) and a similar reasoning