Find all points of discontinuity of f, where f is defined by
\[f(x) = \begin{cases} x^{10}-1, & \quad \text{if } x{\leq 1}\\ x^2, & \quad \text{if } x \text{>1} \end{cases}\]
Find all points of discontinuity of f, where f is defined by
\[f(x) = \begin{cases} x^{10}-1, & \quad \text{if } x{\leq 1}\\ x^2, & \quad \text{if } x \text{>1} \end{cases}\]
Solution and Explanation
\(f(x) = \begin{cases} x^{10}-1, & \quad \text{if } x{\leq 1}\\ x^2, & \quad \text{if } x \text{>1} \end{cases}\)
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) = c10-1 and \(\lim\limits_{x \to c}\) f(x) = \(\lim\limits_{x \to c}\) f(x10-1) = c10-1
∴\(\lim\limits_{x \to c}\) 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
\(\lim\limits_{x \to 1^-}\) f(x) =\(\lim\limits_{x \to 1^-}\)(x10-1)=1-1=0
The right hand limit of f at x = 1 is,
\(\lim\limits_{x \to 1^+}\) f(x) = \(\lim\limits_{x \to 1^+}\)(x2) = 12 = 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):
Ifc>1, then f(c) = c2
\(\lim\limits_{x \to c}\) f(x) = \(\lim\limits_{x \to c}\) (x2) = c2
∴\(\lim\limits_{x \to c}\) 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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Notes on Continuity and differentiability
Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.