Answer to Question #291945 in Real Analysis for Reens

Question

Prove that continuous function of a continuous function is continuous.

Answer 1

let E>0 be given

take E/2 as epsilon for both f and g,

since f,g are continuous we have

|f(x)-f(a)|< E/2, for all |x-a|<d1

|g(x)-g(a)|<E/2, for all |x-a|< d2

let d = min{d1,d2}

for |x-a|< d, we have

|f(x)-f(a)|< E/2,

|g(x)-g(a)|< E/2,

adding we get

|f(x)-f(a)|+|g(x)-g(a)|< E

we know that |f(x)-f(a) +g(x)-g(a)|<|f(x)-f(a)|+|g(x)-g(a)| < E

thus proved