Let $\displaystyle f(x)$ be defined on an interval $\displaystyle I$ (not assumed compact), and assume that its secants have bounded slope, i.e., for any two distinct points on the graph of $\displaystyle f(x)$ over $\displaystyle I,$ the slope $\displaystyle \lambda$ of the line joining them is bounded: $\displaystyle |\lambda| < K,$ where $\displaystyle K$ is some fixed number not depending on the two points selected.

$\displaystyle (a)$ Prove $\displaystyle f(x)$ is uniformly continuous on $\displaystyle I.$

$\displaystyle (b)$ Does $\displaystyle \sqrt{x}$ on $\displaystyle [0,1]$ satisye the above hypothesis on $\displaystyle f(x)?$ Is it uniformly continuous on the interval?