Find all points of discontinuity of f,where f is defined by
\[f(x) = \begin{cases} \frac {x}{|x|}, & \quad \text{if } x \text{<0}\\ -1, & \quad \text{if } x {\geq 0} \end{cases}\]
Find all points of discontinuity of f,where f is defined by
\[f(x) = \begin{cases} \frac {x}{|x|}, & \quad \text{if } x \text{<0}\\ -1, & \quad \text{if } x {\geq 0} \end{cases}\]
Solution and Explanation
\(f(x) = \begin{cases} \frac {x}{|x|}, & \quad \text{if } x \text{<0}\\ -1, & \quad \text{if } x {\geq 0} \end{cases}\)
It is known that, x<0\(\implies\)|x| = -x
Therefore, the given function can be rewritten as
\(f(x) = \begin{cases} \frac {x}{|x|}=\frac {x}{-x}=-1, & \quad \text{if } x \text{<0}\\ -1, & \quad \text{if } x {\geq 0} \end{cases}\)
\(\implies\)f(x) = -1 for all x∈R
Let c be a point on the real number.Then, \(\lim\limits_{x \to c}\) f(x) = \(\lim\limits_{x \to c}\) (-1) = -1
Also, f(c) = -1 = \(\lim\limits_{x \to c}\) f(x)
Therefore ,the given function is a continuous function.
Hence, the given function has no point of discontinuity.
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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.