Limit of Riemann-Type Sum Partition

Let $f : [0, 1] \to (0, \infty)$ be a continuous function. For $n \in \mathbb{N}$, $n \ge 2$, consider $0 = t_0 < t_1 < \dots < t_n = 1$, such that $$ \int_{t_0}^{t_1} f(t) \, dt = \int_{t_1}^{t_2} f(t) \, dt = \dots = \int_{t_{n-1}}^{t_n} f(t) \, dt. $$ Compute $$ \lim_{n \to \infty} \frac{n}{\frac{1}{f(t_1)} + \frac{1}{f(t_2)} + \dots + \frac{1}{f(t_n)}}. $$