Check whether the set of integers is countable or not.
A set is said to be countable if its elements can be put into one-to-one correspondence with the set of natural numbers. The set of integers consists of all positive integers, negative integers, and zero. ___ ______ _______ ____ _________.
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दिखाइए कि फलन , जो
द्वारा परिभाषित है, अपने प्रांत पर सतत है, परंतु वहाँ समान रूप से सतत नहीं है। क्या यह फलन किसी भी निश्चित c > 0 के लिए अंतराल
पर एकसमान सतत होगा?
Find the radius of convergence of the series , where
.
Examine the type of discontinuity of the function at where
मान लीजिए कि f एक अवकलनीय फलन है, जिसका अवकलज अंतराल [a, b] पर शून्य नहीं होता। तब दिखाइए कि f या तो सर्वथा वर्धमान है अथवा सर्वथा ह्रासमान है।
State Cauchy’s second theorem on limits. Use it to show that:
Check whether the following sets are open or closed or neither:
(i)
(ii)
State whether the following statements are true or false. Justify your answers with a short proof or a counterexample.
i) The negation of is
.
ii) is countable.
iii) The equation has a real root between -2 and 1.
iv) Every increasing sequence has a convergent subsequence.
v) The function defined as
has a removable discontinuity.
vi) An integrable function can have finitely many points of discontinuity.
vii) The series is divergent.
viii) The function f defined by , has local maximum at
.
ix) Every constant function on [a, b] is Riemann Integrable.
x) The function defined by
is differentiable at
.