On p. 146 in Evans' book Partial Differential equations (AMS, 1998), it is written:
Now for each $ x ∈ \mathbb{R}$ and $t > 0$, define the point $y(x,t)$ to equal the smallest of those points $y$ giving the minimum of $t L(\frac{x−y}{t})+h(y)$. Then the mapping $x → y(x,t)$ is nondecreasing and is thus continuous for all but at most countably many $x$. At a point $x$ of continuity of $y(·,t)$, $y(x,t)$ is the unique value of $y$ yielding the minimum.
I am trying to see why at a point $x$ of continuity of $y(·,t)$, $y(x,t)$ is the unique value of $y$ yielding the minimum. Can someone help me with that?