I need to find a continuous function with its domain in the closed continuous interval $ [0-\frac{\pi}{2}] $. Its complete range or codomain is within closed continuous interval $ [1-\sqrt2] $. I also know that from 0 to $ \frac{\pi}{4} $ my function behaves exactly like a $ \frac{1}{\cos(x)} $ function, while beyond $ \frac{\pi}{4} $ till $ \frac{\pi}{2} $ it behaves like a $ \frac{1}{\sin(x)} $ function. Is there such a continuous function?
