Which of the following statements is FALSE?
A) All polynomial functions are continuous everywhere.
B) The absolute value function f(x) = l x l is continuous everywhere
C) All rational functions are continuous everywhere
D) Square root function is always continuous at every number in its domain.
The correct answer and explanation is :
The false statement is:
C) All rational functions are continuous everywhere.
Explanation:
A function is continuous at a point if there are no breaks, jumps, or asymptotes at that point. Now, let’s analyze each statement carefully:
- Statement A (True): Polynomial functions (e.g., ( f(x) = x^2 + 3x – 5 ), ( g(x) = x^5 – 2x^3 + 4 )) are continuous everywhere because they are built from sums, products, and powers of ( x ), which are always continuous. There are no restrictions or undefined points in polynomial functions.
- Statement B (True): The absolute value function, ( f(x) = |x| ), is continuous everywhere. Even though it has a sharp corner at ( x = 0 ), it does not have any jumps or breaks, meaning the left-hand and right-hand limits exist and are equal to the function value.
- Statement C (False): Rational functions are not always continuous everywhere because they have denominators that can be zero, leading to undefined points (vertical asymptotes). A rational function is a ratio of two polynomials:
[
R(x) = \frac{P(x)}{Q(x)}
]
It is continuous except where ( Q(x) = 0 ), which creates discontinuities (holes or vertical asymptotes). For example,
[
f(x) = \frac{1}{x-3}
]
is undefined at ( x = 3 ), making it discontinuous there. - Statement D (True): The square root function, ( f(x) = \sqrt{x} ), is continuous for all values in its domain (i.e., ( x \geq 0 )). There are no jumps or breaks in its valid range.
Thus, the incorrect (false) statement is C, because rational functions can have points of discontinuity.