If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

If f is not a function, but is instead a partial function, it is called a partial operation.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1 / g ′( f ( 0 )).

If k, m, and n are 1, so that and, then the Jacobian matrices of f and g are.

If f is a function of as above, then the second derivative of is:

If we let f be a function

If a probability distribution has a density function f ( x ), then the mean is

If the function f is not linear ( i. e. its graph is not a straight line ), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

If the limit exists, then f is differentiable at a.

If f is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from.

If the limit exists, meaning that there is a way of choosing a value for Q ( 0 ) that makes the graph of Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q ( 0 ).

If y = f ( x ) is differentiable at a, then f must also be continuous at a.

If in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism.

If f is a surjection and a ~ b ↔ f ( a ) = f ( b ), then g is a bijection.

If m and n are natural numbers and f ( x ) is a smooth ( meaning: sufficiently often differentiable ) function defined for all real numbers x in the interval, then the integral

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If he sees the heroic in a Sartoris or a Sutpen, he sees also -- and he shows -- the blind and the mean, and he sees the Compson family disintegrating from within.

If Robinson was a liar and a slanderer, he was also a very canny gentleman, for nothing that Pike could do would pry so much as a single word out of him.

If, the editors sometimes, dozed and printed pretentious, New, York-mind, dross, they, also printed, Malraux,, Silone,, Chiaromonte,, Gide, Bellow,, Robert Lowell, Francis Fergusson, Mary McCarthy, Delmore Schwartz, Mailer, Elizabeth Hardwick, Eleanor Clark,, and a host of, other good writers.

If they also defeat the school bill, the GOP task force won't have much research to do.

If you are a nonresident alien and a resident of Puerto Rico, your return is also due June 15, 1962, or the 15th day of the 6th month after the close of your fiscal year.

If your child is under 19 or is a student you may also claim an exemption for him if he qualifies as your dependent, even though he earns $600 or more.

If the superintendents do not receive more cooperation from Handlers, it has been suggested that licensed Judges also be qualified to judge this Class.

If the mechanics are similar, we can also infer that the thermodynamics will also be similar.

If word classes differ in their resistance or liability to stem replacement within meaning slot, it is conceivable that individual meanings also differ with fair consistence trans-lingually.

If Cynewulf was literate, the Beowulf poet may have been also, and so may the final redactor of The Iliad and The Odyssey.

If high accuracy is required in preflight leveling, it is usually necessary to integrate or doubly integrate the accelerometer outputs ( this also minimizes the noise problem ).

If the farm is in an industrial area where wages are high, farm labor costs will also be high.

If one takes the middle number, 5, and multiplies it by 3 ( the base number of the magic square of three ), the result is 15, which is also the constant sum of all the rows, columns, and two main diagonals.

If you know that he ( God ) is just ( righteous ), know that everyone also who does what is just ( righteous ) has been born of him.

If it was intentional, it was now also unnecessary ; ;

If an affidavit is notarized or authenticated, it will also include a caption with a venue and title in reference to judicial proceedings.

If one also removes the second postulate (" a line can be extended indefinitely ") then elliptic geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.

* If S and T are in M then so are S ∪ T and S ∩ T, and also a ( S ∪ T )

* If a set S is in M and S is congruent to T then T is also in M and a ( S )

If there are proteins left in the shell matrix, it is also possible that they can trigger an allergic ( asthmatic ) attack.

If necessary, motoring slowly around the location of the anchor also helps dislodge it.

If the Onemen Bay is considered part of the river, it also receives the Velikaya River ( Chukotka ) from the south and the Kanchalan River from the north.

If this etymology is combined with the tradition reported by Geoffrey of Monmouth stating that Ambrosius Aurelianus ordered the building of Stonehenge – which is located within the parish of Amesbury ( and where Ambrosius was supposedly buried ) – and with the presence of an Iron Age hill fort also in that parish, then it may be tempting to connect Ambrosius with Amesbury.

If an alpha channel is used in an image, it is common to also multiply the color by the alpha value, to save on additional multiplications during compositing.