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 f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.

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 above errors be eliminated, the two astigmatic surfaces united, and a sharp image obtained with a wide aperture — there remains the necessity to correct the curvature of the image surface, especially when the image is to be received upon a plane surface, e. g. in photography.

If, in an unsharp image, a patch of light corresponds to an object point, the center of gravity of the patch may be regarded as the image point, this being the point where the plane receiving the image, e. g., a focusing screen, intersects the ray passing through the middle of the stop.

If the creditworthiness of the issuer deteriorates ( e. g. rating downgrade ) and its credit spread widens, the bond price tends to move lower, but, in many cases, the call option part of the convertible bond moves higher ( since credit spread correlates with volatility ).

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

If there are g ( E ) dE states with energy E to E + dE, then the Boltzmann distribution predicts a probability distribution for the energy:

If BCG is accidentally given to an immunocompromised patient ( e. g., an infant with SCID ), it can cause disseminated or life-threatening infection.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

If matrix support is polar ( e. g. paper, silica etc.

If the disk was not otherwise prepared with a custom format, ( e. g. for data disks ), 664 blocks would be free after formatting, giving 664 × 254 = 168, 656 bytes ( or almost 165 kB ) for user data.

If G is a group, and g is a fixed element of G, then the conjugation map

If is a constant, the solution is particularly simple, and describes, e. g., if, the exponential decay of radioactive material at the macroscopic level.

Many results about plane figures are proved, e. g., If a triangle has two equal angles, then the sides subtended by the angles are equal.

If the gain is not constant, e. g., by clipping the output signal at the limits of its capabilities, the output signal will be distorted.

If we allow also the system boundary to move ( e. g. due to moving pistons ) we get a rather general form of the first law for open systems.

If there is another functional group at a carbon, it may be named with the Greek letter, e. g., the gamma-amine in gamma-aminobutanoic acid is on the third carbon of the carbon chain attached to the carboxylic acid group.

If f: A < sub > 1 </ sub > → A < sub > 2 </ sub > and g: B < sub > 1 </ sub > → B < sub > 2 </ sub > are morphisms in Ab, then the group homomorphism Hom ( f, g ): Hom ( A < sub > 2 </ sub >, B < sub > 1 </ sub >) → Hom ( A < sub > 1 </ sub >, B < sub > 2 </ sub >) is given by φ g o φ o f. See Hom functor.

If f: X < sub > 1 </ sub > → X < sub > 2 </ sub > and g: Y < sub > 1 </ sub > → Y < sub > 2 </ sub > are morphisms in C, then the group homomorphism Hom ( f, g ): Hom ( X < sub > 2 </ sub >, Y < sub > 1 </ sub >) → Hom ( X < sub > 1 </ sub >, Y < sub > 2 </ sub >) is given by φ g o φ o f.

If it were not for an old professor who made me read the classics I would have been stymied on what to do, and now I understand why they are classics ; ;

If I even hint at it do you think it will matter that you are his nephew -- and not even a blood nephew ''??

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

Defoe then commented, `` If they Could Draw that young Gentleman into Their Measures They would show themselves quickly, for they are not asham'd to Say They want only a head to Make a beginning ''.

If Jews are identified as a religious body in a controversy that comes before a national or international tribunal, it is obviously compatible with the goal of human dignity to protect freedom of worship.

If they are right, they will prevail of and by themselves.

Without preliminaries, Esther asked him, `` If you are a world citizen, will you take Garry Davis' place in his tent while he goes to the hospital ''??

If we are to believe the list of titles printed in Malraux's latest book, La Metamorphose Des Dieux, Vol. 1 ( ( 1957 ), he is still engaged in writing a large novel under his original title.

If the would-be joiner asks these questions he is not likely to be duped by extremists who are seeking to capitalize on the confusions and the patriotic apprehensions of Americans in a troubled time.

If we break the minister to our bit, we are buying back our own sins.

If the record buyer's tastes are somewhat eclectic or even the slightest bit esoteric, he will find them satisfied on educational records.

If Daddy's books are out of bounds his own picture books are not.

If it will simply delay the debates until the qualifications are closed next spring, and then carry all the candidates on a tour of debates, it can provide a service to the state.

If they are to be commended for foresight in their planning, what then is the judgment of a town council that compounds this problem during the planning stage??

If the Communists are sincere in wanting a united, neutral and disarmed Germany, it might well be advantageous for the German people in this nuclear age.

If only this could be done more often -- with such heartening results -- many of the earth's `` big problems '' would shrink to the insignificances they really are.

If any are left, presently, we may expect to see signs specifically prohibiting the feeding of them too.

If the raw population figures are crucially relevant, then it is idle to think of liberation, as idle as to suppose that Poland might liberate Russia.

If it is not enough that all of our internationalist One Worlders are advocating that we join this market, I refer you to an article in the New York Times' magazine section ( Nov. 12, 1961 ), by Mr. Eric Johnston, entitled `` We Must Join The Common Market ''.

If the UN troops are not mercenaries then the Hessians were not mercenaries either.