Prove that the function f defined by

is discontinuous, ∀ x ∈ , using the sequential definition of continuity.

Prove that the union of two closed sets is a closed set. Give an example to show that union of an infinite number of closed sets need not be a closed set.

Check whether the function f given by: 

has local maxima and local minima.

Show that there is no real number, k for which the equation, x⁴ − 3x² + k = 0 has two distinct roots in the interval [2,3].

Test the series:

for absolute and conditional convergence.

Examine the function f :   defined by

for continuity on . If it is not continuous at any point of , find the nature of discontinuity there.