# Space of all continuous real valued functions on $[0,1]$ with sup metric is path connected

How can I prove that the function space $\mathcal{C}[0,1]$ of all continuous real valued functions on $[0,1]$ with the sup metric is connected?

I think the sup metric is as follows:

If $f, g$ are in $\mathcal C [0,1]$, then $$d(f,g)= \sup_{x\in [a,b]} |f(x)-g(x)|$$

To show that it's connected, I think we can better prove that it's path connected, which implies connectedness.

Hint: To show that the space is path connected, we need some continuous $\gamma : [0,1] \rightarrow \mathcal{C}[0,1]$ such that $\gamma(0)=f(x), \gamma(1) = g(x)$. (Since $\forall t \in [0,1]$, $\gamma(t)$ is a continuous function, denote $h_t(x)$ to be the continuous function given by $\gamma(t)$.)
Define $\gamma(t) =h_t(x) = f(x) + t(g(x)-f(x)).$ It is easily shown that this is continuous (in fact, uniformly continuous).