Question

Discuss the continuity of the following functions. (a) f ( x ) = sin x + cos x (b) f ( x ) = sin x − cos x (c) f ( x ) = sin x × cos x

Answer 1

Let g(x)=sinx.

Let k be any real number.

At x=k, the given function becomes,

g( k )=sink.

Consider the left hand limit,

LHL= lim x→ k − g( x ) lim x→ k − sinx= lim h→0 sin( k−h ) = lim h→0 sinkcosh−cosksinh =sink

Consider the right hand limit,

RHL= lim x→ k + g( x ) lim x→ k + sinx= lim h→0 sin( k+h ) = lim h→0 sinkcosh+cosksinh =sink

Here, at x=k, LHL=RHL.

Thus, the function is continuous for all real numbers.

Now, let h(x)=cosx.

Let k be any real number.

At x=k, h( k )=cosk.

Consider the left hand limit,

LHL= lim x→ k − h( x ) lim x→ k − cosx= lim h→0 cos( k−h ) = lim h→0 coskcosh+sinksinh =cosk

Consider the right hand limit,

RHL= lim x→ k + h( x ) lim x→ k + cosx= lim h→0 cos( k+h ) = lim h→0 coskcosh−sinksinh =cosk

Here at x=k, LHL=RHL.

Hence, the function is continuous for all real numbers.

If g and h are two continuous functions, then the functions g+h, g−h and gh are also continuous functions.

(a)

The function is given as f( x )=sinx+cosx. Since f( x )=g+h, therefore the function is continuous.

(b)

The function is given as f( x )=sinx−cosx. Since f( x )=g−h, therefore the function is continuous.

(c)

The function is given as f( x )=sinx⋅cosx. Since f( x )=g⋅h, therefore the function is continuous.

Thus, f( x )=sinx+cosx, f( x )=sinx−cosx and f( x )=sinx⋅cosx are continuous functions.