$$ K = \{x | f_x(x)↓ \} $$ I need to show that: $$ K \leq_m \{x | f_x(c)↓ \} $$(for a given number c). I need a total-computable function h, such that: $$ x \in K \iff h(x) \in \{x | f_x(c)↓\} $$
I need to show it using the s-n-m theorem. My attempt was:
$$\text{let t be such that: } f_t = id.$$ $$\text{and let p be such that: } f_p = \emptyset$$ then $$h(x) = t \text{(if f_x(x)↓), p (otherwise)}$$
Any tips why my is wrong and how can I do it with the s-n-m theorem?