In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More sufficiently small changes in the input of a continuous function result in arbitrarily small changes in its output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the core concepts of topology, treated in full generality below; the introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. A stronger form of continuity is uniform continuity. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist; as an example, consider the function h, which describes the height of a growing flower at time t.

This function is continuous. By contrast, if M denotes the amount of money in a bank account at time t the function jumps at each point in time when money is deposited or withdrawn, so the function M is discontinuous. A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of y = f as follows: an infinitely small increment α of the independent variable x always produces an infinitely small change f − f of the dependent variable y. Cauchy defined infinitely small quantities in terms of variable quantities, his definition of continuity parallels the infinitesimal definition used today; the formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasn't published until the 1930s. Like Bolzano, Karl Weierstrass denied continuity of a function at a point c unless it was defined at and on both sides of c, but Édouard Goursat allowed the function to be defined only at and on one side of c, Camille Jordan allowed it if the function was defined only at c.

All three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854. A real function, a function from real numbers to real numbers can be represented by a graph in the Cartesian plane. A more mathematically rigorous definition is given below. A rigorous definition of continuity of real functions is given in a first course in calculus in terms of the idea of a limit. First, a function f with variable x is said to be continuous at the point c on the real line, if the limit of f, as x approaches that point c, is equal to the value f. A function is said to be discontinuous at some point; these points themselves are addressed as discontinuities. There are several different definitions of continuity of a function. Sometimes a function is said to be continuous. In this case, the function f = tan, with the domain of all real x ≠ π/2, n any integer, is continuous.

Sometimes an exception is made for boundaries of the domain. For example, the graph of the function f = √x, with the domain of all non-negative reals, has a left-hand endpoint. In this case only the limit from the right is required to equal the value of the function. Under this definition f is continuous at the boundary x = 0 and so for all non-negative arguments; the most common and restrictive definition is that a function is continuous if it is continuous at all real numbers. In this case, the previous two examples are not continuous, but every polynomial function is continuous, as are the sine and exponential functions. Care should be exercised in using the word continuous, so that it is clear from the context which meaning of the word is intended. Using mathematical notation, there are several ways to define continuous functions in each of the three senses mentioned above. Let f: D → R be a function defined on a subset D of the set R of real numbers; this subset D is the domain of f. Some possible choices include D = R, or, for a and b real numbers, D = =, or D = =

Tecrea Ltd is a biotech company located in the London Bioscience Innovation Centre. The company is involved in research and development to improve the cell and tissue delivery of drugs and reagents; the approach involves nanomedicine. Tecrea along with Cobra Biologics was awarded the Innovate UK Grant worth £112K to Develop AAV Scalable Production Bioprocess Founded in 2012, as a spinout from the Royal Veterinary College, University of London, the company has developed a line of tools aimed at improving scientific experiments through enhanced delivery of a range of molecules into cells. Tecrea Research Reagents and Drug Formulation products HappyFect When mixed with DNA or RNA, HappyFect forms nanoparticles that enable efficient transfection with low toxicity There are two types of HappyFect:HappyFect-PLASMID HappyFect-RNAi NanocargoTecrea’s NanoCargo products are aimed at improving the delivery of small molecules and proteins, including antibodies, into a range of cell types and tissues. Nanocargo can deliver cargo molecules into gram negative and gram positive bacteria.

There are two types of Nanocargo:Nanocargo-PRO Nanocargo-SM Antimicrobial - SanZone SanZone is used to ensure that incubators and water baths remain clean and infection-free. When culturing cells of any type, it is imperative that incubators be sterile areas; the presence of SanZone kills microbes and prevents infection from accumulating in water baths and incubators. Nanocin Nanocin is a new platform for nanoparticle-based delivery of small molecules that can be applied to small molecule reagents and diagnostic probes. Official website

Omar Akram is a Grammy Award winning recording artist, producer and pianist. In 2013, he became the first Afghan-American to win a Grammy Award for Best New Age Album for his fourth studio album, Echoes of Love, he is an inspirational writer who contributes to The Huffington Post. Akram was born in New York City, he grew up traveling around the world as the son of a United Nations diplomat living everywhere from Prague to Havana as well as his ancestral home of Afghanistan. He notably met Fidel Castro at the age of 14, said to have allowed him to sneak into the local Cuban jazz clubs, the sounds of which influenced his formative compositional style. In 1993, Akram moved to Los Angeles where he performed in various Mainstream Top 40 bands while writing his own music. In 2002, he signed a recording deal with Real Music, his first commercial album release that same year, titled Opal Fire, reached Billboard's New Age Top 15 Chart that same year. His sophomore release Free As A Bird in 2004, featured world class violinist Charlie Bisharat and Grammy Award-winning saxophonist Eric Marienthal.

This album reached Billboard's New Age Top 15 Chart that same year. In 2007, Secret Journey critically acclaimed, featured Ardeshir Farah. In 2012, Akram was nominated and won the Grammy Award for Best New Age Album for his album Echoes of Love; this album was followed by Daytime Dreamer in 2013 which featured six new recordings along with tracks released from Opal Fire and Free As A Bird. Akram's most recent album release "Destiny," co-produced with producer Walter Afanasieff, was independently released through his own company, Twinbrook Entertainment, on August 9, 2019. Afanasieff contributes vocals to the first single release from the album, "Here I Am". An accompanying official music video was directed by Erik White. Shardad Rohani conducted the Slovak Radio Symphony Orchestra on two tracks; the album was mixed by four-time Grammy winner Dave Reitzas at Westlake Studios in Hollywood, CA. Akram is producing a docu-series titled "Omar's Music Chamber." Grammy Award for Best New Age Album for Echoes of Love in 2014.

Omar resides in Los Angeles, California with his wife and daughter. Real Music Real Music Real Music. Secret Journey peaked at #12 on Billboard's New Age Chart. Real Music. Echoes Of Love was awarded a Grammy for Best New Age Album. Real Music Twinbrook Entertainment