Find all points of discontinuity of f,where f is defined by
\[f(x) = \begin{cases} \frac {|x|} {x}, & \quad \text{if } x { \neq 0}\\ 0, & \quad \text{if } x { =0} \end{cases}\]
Find all points of discontinuity of f,where f is defined by
\[f(x) = \begin{cases} \frac {|x|} {x}, & \quad \text{if } x { \neq 0}\\ 0, & \quad \text{if } x { =0} \end{cases}\]
Solution and Explanation
\(f(x) = \begin{cases} \frac {|x|} {x}, & \quad \text{if } x { \neq 0}\\ 0, & \quad \text{if } x { =0} \end{cases}\)
It is known that, x<0 \(\implies\) |x| = -x and x>0 \(\implies\) |x| = 0
Therefore,the given function can be rewritten as
\(f(x) = \begin{cases} \frac {|x|} {x}=\frac {-x}{x}=-1, & \quad \text{if } x { <0}\\ 0, & \quad \text{if } x { =0} \\ \frac {|x|} {x}=\frac {x}{x}=1, & \quad \text{if } n \text{ >0} \end{cases}\)
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<0,then f(c) = -1
\(\lim\limits_{x \to c}\) f(x) = \(\lim\limits_{x \to c}\) (-1) = -1
∴\(\lim\limits_{x \to c}\) f(x) = f(c)
Therefore, f is continuous at all points x, such that x<0
Case (ii):
If c = 0,then the left hand limit of f at x = 0 is,
\(\lim\limits_{x \to 0^-}\) f(x) = \(\lim\limits_{x \to 0^-}\) (-1)=-1
The right hand limit of f at x = 0 is,
\(\lim\limits_{x \to 0^+}\) f(x) = \(\lim\limits_{x \to 0^+}\) (1) = 1
It is observed that the left and right hand limit of f at x = 0 do not coincide.
Therefore,f is not continuous at x = 0
Case(iii):
Ifc>0, then f(c) = 1
\(\lim\limits_{x \to c}\) f(x) = \(\lim\limits_{x \to c}\) (1) =1
∴\(\lim\limits_{x \to c}\) f(x) = f(c)
Therefore, f is continuous at all points x, such that x>0
Hence,x = 0 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.