The mountain climbing theorem is a synchronization result for two continuous motions along a function graph, with the two moving points kept at the same height (Whittaker 1966, Numberphile 2026).
In one form, let ![f:[0,1]->[0,infty)](/images/equations/MountainClimbingTheorem/Inline1.svg)
be a continuous
function with 
. Suppose there are points 
,
where 
is a positive integer, such that 
is a monotonic function
on each closed interval ![[x_(j-1),x_j]](/images/equations/MountainClimbingTheorem/Inline6.svg)
. A restriction may be a constant
function. Then there exist continuous functions ![u,v:[0,1]->[0,1]](/images/equations/MountainClimbingTheorem/Inline7.svg)
with 
and 
such that
 |
for every ![t in [0,1]](/images/equations/MountainClimbingTheorem/Inline10.svg)
(Whittaker 1966). Either motion may pause or reverse direction. The intermediate
value theorem applied to 
shows that the moving points meet.
The restriction on the sign of 
matters for this conclusion. For example, 
satisfies the conditions at the endpoints
but does not admit such a pair of motions (Whittaker 1966).
