إلى استاذنا القدير Uaemath

بصراحة عندي أسئلة وودي إنك تساعدني فيها..

و هي في التحليل الحقيقي

إذا ممكن ...

 

 

Sec :Differentiation

1-let g :R goes to R be defined by
g(x):X+(2X^2)(sin(1/x)for x does not equal 0
and g(0):=0.show that g'(x)=1,but in every neighborhood of 0
the derivative g'(x) takes on both positive
and negative values. thus g is not monotonic in any neighborhood of 0


The second Question:

Sec: the Derivative

1-If r>0 is a rational number,let be defined by
f(x) =(X^r) (sin(1/x)) for x does not equal 0
and f(x):=0 Determine those value of r for which
f '(0) exists.



The third question:

Sec : Continuous Function

A function f: R goes to R is said to be Periodic on R
if there exists a number P>0 such that
f(x + p) = f(x) for all x include R
.
Prove that a continuous periodic function on R
is bounded and uniformly continuous on R