I have a piecewise linear interpolation:
$$ B(t) = \frac{t_{l+1}-t}{t_{l+1}-{t_{l}}} B_l + \frac{t-t_l}{t_{l+1}-{t_{l}}} B_{l+1} \quad \text{ if $t \in (t_l, t_{l+1})$;}$$ $B(t_l)=B_{l}$ and $B(t) = B_0$ for $t \leq0$.
I want to use that $B(t)$ is uniformly continuous on whole $R$. Is there an easy argument for that? (For example if from continuity on R and uniform continuity everywhere but in $t_1, t_2, \cdot$ (countable many points) would follow uniform continuity everywhere). Or maybe there is an elegant way via Lipschitz-continuity? I think the uniform continuity inside the intervals $(t_l, t_{l+1})$ is clear, it's just the question how to put it all together without having to recede to the $\epsilon$-$\delta$-definition.
Q: Is there a better (shorter/more elegant) way to follow uniform continuity here?
The red line would be a line with slope = supremum of the slopes on the distinct intervals, the orange line is a sample secant line.
Your approach via Lipschitz continuiuty may work: The piecewise linear interpolation is Lipschitz continuous with Lipschitz constant equal to the largest slope within the interpolation intervals - provided this exists. Hence, your function is Lipschitz continuous if the slopes on the interpolation intervals are uniformly bounded. Otherwise, your function will not even be uniformly continuous.