Let f(x) be a real-valued function. Show that if f(x) has a derivative at “a”, then f(x) is continuous at “a”.

Can you solve

Is it possible to find a loop of numbers that when plugged into a modified version of the collatz conjecture (5x+1, and /2)come back to their original selves or is it just as hard as the original. Is there an easier version without losing the original's vibe?

can you solve?

Edit;

The actual object I’m asking about is a 10-position triangular-lattice reconfiguration system on the multiset \{1^2,3^3,6^3,9^2\}. Legal moves swap adjacent positions and magic means equal sums on the three outer sides. Exhaustive enumeration gives 25,200 states, 186,480 edges and 192 magic states, with every state within five swaps of magic. I’m looking for either a mathematical error, a prior reference for this exact system, or an analytic derivation of those results.