Which of the following functions is not continuous at the origin?

Concept:

• A function is said to continuous at a given value if the function does not have a sudden change at that value.
• If two functions are individually continuous then it’s summation difference, the product will be also continuous.
• A function is continuous at x = a
  If L.H.L = R.H.L = f(a)
  i.e. $\underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}f\left(x\right)=f\left(a\right)$ 
• If f(x) and g(x) are individually continuous then it’s f(x) ± g(x), f(x) ⋅ g(x) and $\frac{f\left(x\right)}{g\left(x\right)};g\left(x\right)\ne 0$ will be continuous.

Calculation:

Option (A):

Let f(x) = sin x

From graph, at origin i.e. x = 0, sin x is continuous.

Option (B):

Let g(x) = x

From graph at origin i.e. x = 0

g(x) = x is continuous.

Option (C):

Let h(x) = x sin x

h(x) = g(x) ⋅ f(x)

Since f(x) and g(x) is individually continuous at origin, so h(x) will be also continuous.

Option (D):

Let $h\left(x\right)=\frac{\sin x}{x}=\frac{f\left(x\right)}{g\left(x\right)}$ 

Since sin x is continuous at origin and x is also continuous at origin and values are zero.

i.e. $h\left(x\right)=\frac{\sin x}{x}$ 

At origin:

$h\left(x=0\right)=\frac{\sin \left(0\right)}{0}=\frac{0}{0}$  is a indeterminate form)

Therefore, we can say that.

h(x) will be discontinuous at origin as at x = 0, h(x) become undefined.

So, option (D) is correct as $\frac{\sin x}{x}$ is discontinuous at origin.