Question

Please Answer 135 Below Completely:

Definition Let E-R and f : E-+ R be a function. For some p E E' we say that f is continuous at p if for any ε > 0, there exists a δ > 0 (which depends on ε) such that for any x E E with |x-Pl < δ we have If(x) -f(p)le KE. This is often called the rigorous δ-ε definition of continuity. A couple of things to note about this definition. First, it is very similar to the definition of convergence for sequences where N corresponds with δ and n > N corresponds with E (p-e, p+ ε) n E. Second, δ is related to distance between points in the dom ain, while ε is related to distance between points in the range. If a function f is continuous for all points pe E, then we say that f is continuous onE 135. Suppose f : A R and g : B R are functions with f(A) C B. Define h : A -> R by h(x) -glf()), in other words, h is the composition of f and g. Suppose for some x E A, f is continuous at r and g is continuous at f(x). Prove that h is also continuous at r.

Answer 1

"Given: f: A rightarrow R g: B rightarrow R f, g are continuous Let f be continuous at p elementof A therefore g is continuous at f (p) elementof B To prove: g (f (x)) = (g compositefunction f) (x) = h (x) is continuous i.e. we have to prove that: forall epsilon > 0 exists some delta > 0 such that: