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