*The full question is: *
If $f:R→R$ is a continuous increasing bijection function and $g:R→R$ is a continuous decreasing bijection function, then prove or disprove that $h(x)=f(x)−g(x)$ is a bijection function.
Am not good at analysis, know of the concept of continuity (from left & right, based on limits) only. This concept helps in deriving properties needed for my basic knowledge.
If two functions are monotonic (added to emphasize further the increasing/decreasing property), bijective (implies to me- that range exists for entire domain, but significance is not intuitive in this context) continuous functions, then even if the domain (& also range) of one is a subset (need not be proper), then still $h(x)$ 'will' not be bijective, as shown below:
If $f$ is suppose increasing, and $g$ decreasing over the same interval (both are having same domain and ranges, with none subset of the other, and are equal); then can imagine (with no value assigned to domain, function, hence to range to substantiate) that one function ($f$) is increasing with same domain and range, while another ($g$) is decreasing over the same domain (with the same range). Hence, the difference is zero, which forms the range of $h(x)$ and is a constant value, and hence is not a bijection function even, as $h(x)$ is having the same value for all elements. In case, the range elements have a constant difference for $f, g$, then still the same situation remains.