(xiii)

A real function f is said to be continuous at x = c, where c is any point in the domain of f if

h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

A function is continuous at x = c if

Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)

For x < –1, f(x) is having a constant value, so the curve is going to be straight line parallel to x–axis.

=>  it is everywhere continuous for x < –1.

Similarly for –1 < x < 1, plot on X–Y plane is a straight line passing through origin.

=>  it is everywhere continuous for –1 < x < 1.

And similarly for x > 1, plot is going to be again a straight line parallel to x–axis

∴ it is also everywhere continuous for x > 1

As x = –1 is a point at which function is changing its nature so we need to check the continuity here.

f (–1) = –2

∴ f (x) is continuous at x = –1

Also at x = 1 function is changing its nature so we need to check the continuity here too.

f (1) = 2 [using equation 1]

∴ f (x) is continuous at x = 1

Thus, f(x) is continuous everywhere and there is no point of discontinuity.