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