Question

Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}x^{10} - 1, & \text{ if }  x \leq 1 \\ x^2 , & \text{ if } x > 1\end{cases}\]

Answer 1

Given: `f(x)[{[x^10- 1 ,  \text{ if } ≤ 1],[x^2,\text{ if }    x>1]]`

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

if `c< 1, ` then `f(c)=c^10-1` and `lim_(x-> c) f(x)=lim_(x->c)(x^10  -1 )=c^10  -1`

`∴ lim_(x->c)f(x)=f(c)`

Therefore, f is continuous at all points x, such that x < 1

Case II:

If c = 1, then the left hand limit of f at x = 1 is,

`lim_(x->c)f(x)=lim_(x->c)(x^10  -1)=1^10  -1 =1-1=0`

The right hand limit of f at x = 1 is,

`lim_(x->1)f(x)=lim_(x->1)(x^2)=1^2=1`

It is observed that the left and right hand limit of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1

Case III:

`if c>1, " then "  f(c)=c^2`

`lim_(x->c)f(x)=lim_(x->c)f(c)=c^2`

`∴lim_(x->c)f(x)=f(c)`

Therefore, f is continuous at all points x, such that x > 1

Thus, from the above observation, it can be concluded that x = 1 is the only point of discontinuity of f.