Thread: Countinuity of Countinued Fraction

Define a function,

the necessary and sufficient conditions for convergence is when,
diverges, thus,
this is geometric.
Diverges when for simplicity let . Now prove that this series is countinous.

I am trying to express transcendental functions in terms of infinite countinous fractions, like the one above. I do not think I will get anywhere

Take the sequence which are continuous and converge pointwise to . Prove it converges uniformly; Then the limit function is also continuous.