Given , `f(x)={{:((1-cos3x)/(x^(2)),"when " xne0),(1,"when "x=0):}`
Prove that f (x) is discontinuous at x = 0 .

A function f (x) is defined as followes: f(x)={{:(xsin.(1)/(x),"when " xne0),(0,"when " x=0):} Show that f (x) is continuous at x =0 .

Let f(x)={{:((1-cos2ax)/(xsin2x),"when " xne0),(" " (1)/(2),"when " x =0):} If f (x) is continuous at x = 0 , find the value of alpha .

Define the continuity of a function at a point within its domain of definition. The definition of a function is given below: f(x)={{:((1-cos5x)/(x^(2))," when "(xne0)),(k," when x = 0"):} Find the value of k for which f(x) is continuous at x = 0.

A function f (x) is defined as follows: f(x)={{:(x+2,"when "xlt2),(x^(2)-1,"when "xge2):} Show that f (x) is discontinuous at x = 2 and the jump of the function at this point is -1 .

If the function f(x)={{:((1-cos(ax))/(x^2)," when "xne0),(1," when "x=0):} be continuous at x=0, then find the value of a.

If f(x)={{:((|sinx|)/(x),"when "xne0),(1,"when "x=0):} then examine the continuity of the function at x = 0.

Let f(x)={{:((1)/(x^(2))sin(x^(2))," when "xne0),(0," when " x=0):} Discuss the continuity of f (x) at x = 0 .

If f(x)={{:(x+sinx,"when "xlt0),(0,"when "xge0):} examine whether f(x) is continuous at x = 0.

Given , f(x)={{:((sin5x)/(3x),"when "xne0),((3)/(5),"when "x=0):} Examine the cotinuity of f (x) at x = 0 .

Prove that, q(x) = {(|x|/x, when x ne 0),(0, when x = 0):} is discontinuous at x = 0.