I need to test if this function $f(x) := \frac{x-2}{3x-4}$ has uniform continuity on given interval $[0,\frac{4}{3})$. I think it isn't uniformly continuous, but I don't know how to prove it.
2026-03-25 04:39:22.1774413562
Proving uniform continuity in $\frac{x-2}{3x-4}$
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Hint: (i) A function continuous on $(a,b) \in \Bbb R$ is uniformly continuous on $(a,b)$ if and only if it can be extended continuously to the end points i.e. it is continuous on $[a,b]$ . (A very well known result, but you can also try to prove it, it's not difficult to prove)
(ii) Your function is not continuous at $x = \frac{4}{3}$ .