1. ## Left continuity

If $\displaystyle f: \mathbb{R} \rightarrow \mathbb{R}$ is increasing and satisfies the conclusion of the intermediate value theorem, then prove f is left continuous.

My attempt:

$\displaystyle \forall r \in [f(a),f(b)] \exists c \in [a,b] \ s.t \ f(c)=r.$

Hence pick $\displaystyle b=a+\epsilon$ and let $\displaystyle \epsilon=\epsilon_n$ s.t $\displaystyle \epsilon_n=1/n$.

Let $\displaystyle r_n=f(a+\epsilon_n)$.

Also let $\displaystyle f(a)=c_1$ (just to make it different from the definition).

therefore $\displaystyle \lim_{n \rightarrow \infty} f(a+\epsilon_n)\rightarrow f(r_{\infty})=f(a)=c_1$

Hence it is left continuous.

Is this right? I have a feeling that i've assumed it to prove it

2. Originally Posted by Showcase_22
If $\displaystyle f: \mathbb{R} \rightarrow \mathbb{R}$ is increasing and satisfies the conclusion of the intermediate value theorem, then prove f is left continuous.

My attempt:

$\displaystyle \forall r \in [f(a),f(b)] \exists c \in [a,b] \ s.t \ f(c)=r.$

Hence pick $\displaystyle b=a+\epsilon$ and let $\displaystyle \epsilon=\epsilon_n$ s.t $\displaystyle \epsilon_n=1/n$.

Let $\displaystyle r_n=f(a+\epsilon_n)$.

Also let $\displaystyle f(a)=c_1$ (just to make it different from the definition).

therefore $\displaystyle \lim_{n \rightarrow \infty} f(a+\epsilon_n)\rightarrow f(r_{\infty})=f(a)=c_1$

Hence it is left continuous.

Is this right? I have a feeling that i've assumed it to prove it
It seems that you should consider $\displaystyle r_n = f(a-\epsilon_n)$ since you want to show left continuity.

3. $\displaystyle \forall r \in [f(a),f(b)] \exists c \in [a,b] \ s.t \ f(c)=r.$

Hence pick $\displaystyle b=a-\epsilon$ and let $\displaystyle \epsilon=\epsilon_n=1/n$.

Let $\displaystyle r_n=f(a+\epsilon_n)$.

Since f is an increasing sequence $\displaystyle r_1>r_2>r_3......$.

f is bounded below by $\displaystyle f(a)$. Therefore $\displaystyle r_{\infty}=f(a)$.

Also let $\displaystyle f(a)=c_1$ (just to make it different from the definition).

therefore $\displaystyle \lim_{n \rightarrow \infty} f(a-\epsilon_n)\rightarrow f(r_{\infty})=f(a)=c_1$

Hence it is left continuous.

Is that better?

4. Originally Posted by Showcase_22
$\displaystyle \forall r \in [f(a),f(b)] \exists c \in [a,b] \ s.t \ f(c)=r.$

Hence pick $\displaystyle b=a-\epsilon$ and let $\displaystyle \epsilon=\epsilon_n=1/n$.

Let $\displaystyle r_n=f(a+\epsilon_n)$.

Since f is an increasing sequence $\displaystyle r_1>r_2>r_3......$.

f is bounded below by $\displaystyle f(a)$. Therefore $\displaystyle r_{\infty}=f(a)$.

Also let $\displaystyle f(a)=c_1$ (just to make it different from the definition).

therefore $\displaystyle \lim_{n \rightarrow \infty} f(a-\epsilon_n)\rightarrow f(r_{\infty})=f(a)=c_1$

Hence it is left continuous.

Is that better?
I would let $\displaystyle r_n = f(a-\epsilon_n)$. Then $\displaystyle r_n$ is an increasing sequence. And so $\displaystyle r_{\infty} = f(a)$. Now taking $\displaystyle r = a, \ c = c_1$ in the Intermediate Value Theorem, we get $\displaystyle f(a) = c_1$. So $\displaystyle \lim_{n \to \infty} f(a-\epsilon_n) = r_{\infty} = f(a) = c_1$.

5. Originally Posted by Showcase_22
If $\displaystyle f: \mathbb{R} \rightarrow \mathbb{R}$ is increasing and satisfies the conclusion of the intermediate value theorem, then prove f is left continuous.

My attempt:

$\displaystyle \forall r \in [f(a),f(b)] \exists c \in [a,b] \ s.t \ f(c)=r.$

Hence pick $\displaystyle b=a+\epsilon$ and let $\displaystyle \epsilon=\epsilon_n$ s.t $\displaystyle \epsilon_n=1/n$.

Let $\displaystyle r_n=f(a+\epsilon_n)$.

Also let $\displaystyle f(a)=c_1$ (just to make it different from the definition).

therefore $\displaystyle \lim_{n \rightarrow \infty} f(a+\epsilon_n)\rightarrow f(r_{\infty})=f(a)=c_1$

Hence it is left continuous.

Is this right? I have a feeling that i've assumed it to prove it

Showcase, the intermediate value theorem says:

If f(x) is continuous in [a,b] and if f(a) =A and f(b)=B,then corresponding to any number C between A and B there exists at least one number c in [a,b] such that f(c) =C.

So it seems to me you are trying to prove the basic assumption of the intermediate value theorem,since if a function is continuous then is left,right continuous