Question

which of the following is false? a) every continuous function is differentiable. b) all rational functions are continuous on their domain. c) a function can have at most two horizontal asymptotes. d) \\(\frac{0}{0}\\) (zero divided by 0) is considered an indeterminate form.

Explanation:

• Option A: A continuous function may not be differentiable. For example, \( y = |x| \) is continuous at \( x = 0 \) but not differentiable there (the graph has a sharp corner). So the statement "Every continuous function is differentiable" is false.
• Option B: Rational functions are continuous on their domain (where the denominator is non - zero) as they are combinations of polynomials, and polynomials are continuous everywhere. So this statement is true.
• Option C: A function can have at most two horizontal asymptotes (one as \( x

ightarrow\infty \) and one as \( x
ightarrow-\infty \), or the same for both, or one of them not existing but at most two distinct ones). This statement is true.

• Option D: \( \frac{0}{0} \) is an indeterminate form (used in limits, like L'Hopital's Rule situations). This statement is true.

Answer:

A. Every continuous function is differentiable.