Barendregt, in his famous book on the lambda calculus has a great justification: once you look at models for the lambda calculus you are forced to introduce not

Barendregt, in his famous book on the lambda calculus has a great justification: once you look at models for the lambda calculus you are forced to introduce notions from analysis, such as continuity. The arguments and proofs would feel right at home in a book on classical analysis. So it's the models that put the calculus in lambda calculus. :) Of course the real answer is probably history, but it's much more interesting to say that you can simply find connections to calculus if you look hard enough...