Question

The function f(x) is said to be piecewise continuous, if it satisfies the following conditions (Dirichlet conditions):
(A) The function must have a finite number of maxima and minima.
(B) The function must have a finite number of infinite discontinuities, in a period of one oscillation.
(C) The function must have an infinite number of maxima and minima.
(D) The function must have a finite number of finite discontinuities, in a period of one oscillation
Choose the correct answer from the options given below

• (A) and (D) only.
• (A) and (C) only.
• (B) and (C) only.
• (B) and (D) only.

Hint:

Piecewise continuity is a crucial concept in mathematical analysis, particularly in Fourier analysis, where it ensures that a function can be represented as a Fourier series.

Answer 1

The Correct Option is A

For a function to be piecewise continuous according to Dirichlet’s conditions, it must have only a finite number of extremities (maxima and minima) and discontinuities within any finite interval. This specifically excludes infinite discontinuities which would make the function not piecewise continuous.