are all rational functions discontinuous?
161135
Rational functions are continuous for all real numbers except at those where the denominator is zero. If the denominator of a rational function f(x) is zero at x=a, then it contains some number of factors of (x - a).
1400747
The left limit does not equal the right limit, and therefore the limit of the difference quotient of f(x) = |x| at x = 0 does not exist. Thus the absolute value function is not differentiable at x = 0.
316647
A rational expression is considered simplified if the numerator and denominator have no factors in common. We can simplify rational expressions in much the same way as we simplify numerical fractions.
318663
If f : X → Y is a continuous function which is 1–1 and onto, then there is a set-theoretic inverse to f, but there are also many examples to show that this inverse function need not be continuous. However, there are numerous conditions of a general nature which imply the continuity of inverses.
509791
Suppose y is a function of x. If this f(x) is a polynomial, then y is a polynomial function. Suppose there is an equation in which left hand side is a polynomial and r.h.s is 0. Then this is a polynomial equation.
44475
A rational expression is proper if the degree of the numerator is less than the degree of the denominator, and improper otherwise. ... In the third example, the numerator and denominator are both fourth-degree polynomials, so the rational function is improper.
437240
The absolute value function is continuous at 0. Since |x| = -x for all x < 0, we find the following left hand limit. Since the two one-sided limits agree, the two sided limit exists and is equal to 0, and we find that the following is true. ... In fact, the absolute value function is continuous (on its entire domain R).
1118073
A function f is one-to-one if for each a and b in the domain of f, if f(a) = f(b) then a = b. Hence if f is an even function and for some number a, a and -a are both in the domain of f then f(a) = f(-a) and yet a ≠ -a and hence f is not one-to-one.
390193
The rational numbers are "closed" under addition, subtraction, and multiplication. ... Irrational numbers are "not closed" under addition, subtraction, multiplication or division.
461386
IN GENERAL , has NO INVERSE , IF IT IS NOT A ONE-TO-ONE FUNCTION.,because only such functions are invertible. ⑵ BUT if a cubic functionis is of the following form/can be converted to the following form, it is invertible : (i) f(x)=(ax+b)³+c, a≠0 , b,c∈|R, with its natural domain, x∈|R or a reduced domain.
181586
The domain of a function f(x) is the set of all values for which the function is defined, and the range of the function is the set of all values that f takes. A rational function is a function of the form f(x)=p(x)q(x) , where p(x) and q(x) are polynomials and q(x)≠0 .
1285691
A function f(x) is even if f(−x)=f(x) and function is odd if f(−x)=−x . Hence sin2(x) is an even function.
1472189
Two important features of any rational function r(x)=p(x)q(x) r ( x ) = p ( x ) q ( x ) are any zeros and vertical asymptotes the function may have. These aspects of a rational function are closely connected to where the numerator and denominator, respectively, are zero.
1087573
If the upper and lower integrals are equal, we say that f is Riemann integrable or integrable. ... Not every bounded function is integrable. For example the function f(x)=1 if x is rational and 0 otherwise is not integrable over any interval [a, b] (Check this).
306364
Quadratics have at most two solutions for every output (dependent variable), but each input (independent variable) only gives one value. The function f(x)=ax2+bx+c is a quadratic function.
570633
Three properties that are universal to all quadratic functions: 1) The graph of a quadratic function is always a parabola that either opens upward or downward (end behavior); 2) The domain of a quadratic function is all real numbers; and 3) The vertex is the lowest point when the parabola opens upwards; while the ...
915578
The rationals, for example, do not form a well-ordering under the usual less-than relation, but there is a way of putting them into one-to-one correspondence with the natural numbers, so it can be well-ordered by the total order implied by this correspondence. Any countable set can be well-ordered.