Continuity‌ ‌and‌ ‌Differentiability‌

Continuity

Continuity of a function is defined as the consistency of a function in a given interval. A function is said to be continuous if at no point, the function breaks or varies from other functional values of the given function. The continuity of a function is defined at every point of the function having the same value. This means that for a real valued function to be continuous, the function at every point in its domain should be continuous.

Continuity of a function f(x) at a point ‘a’ is expressed as

lim_x→a f(x) = f(a)

For example, consider the following function

f(x) = (x + 1)/2 if x is odd

and

f(x) = x/2 if x is even

The function f(x) will be continuous only if it is continuous for both odd and even values of x.

Conditions for continuity of a function

The following conditions are satisfied if a function is continuous:

A function which is defined for an open interval (a,b) will be continuous if at any point, say a, the function is continuous in the interval (a,b).
A function which is defined for a closed interval [a,b] will be continuous if at any point the function is continuous in the intervals (a,b) and [a,b].
If a function f(x) is continuous at a point a then its left hand limit (LHL) and right hand limit (RHL) will be equal to each other and the function at the point a.
lim_x→a- f(x) = lim _x→a f(x) = lim _x→a+ f(x)
If there be two continuous functions f(x) and g(x) then
1. f(x) + g(x) is also continuous
2. f(x) – g(x) is also continuous
3. f(x) . g(x) is also continuous
4. f(x)/g(x) is also continuous for g(x) not equal to 0 at every point.

Differentiability

Differentiability of a function at any point is defined there exists the derivative of the given function at that point. Mathematically, differentiability for a function f(x) is

Lim_x→0 f’(x) = [f(x + h) – f(x)]/h

Where f’(x) is the derivative of f(x). It can also be denoted as d/dx . f(x). The method of obtaining the derivative of a function is called differentiation with respect to a variable. Here, the function f(x) is differentiated with respect to ‘x’.

Rules of differentiation

For a function xⁿ
f(x) = xⁿ

f’(x) = n . x^n-1
is the required derivative.
Derivatives of standard trigonometric functions are
1. d/dx . sinx = cosx
2. d/dx . cosx = - sinx
3. d/dx . tanx = sec²x
4. d/dx . cotx = -cosec²x
5. d/dx . secx = secx . tanx
6. d/dx . cosecx = - (cotx . cosecx)
For functions u and v
(u + v)’ = u’ + v’
For functions u and v
(u - v)’ = u’ - v’
For functions u and v
(u . v)’ = u’ . v + u . v’
For functions u and v
(u/v)’ = [(u’ . v) – (u . v’)]/v²
Where v is a not equal to 0.

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Asexual Reproduction Animals Autotrophic Nutrition Biodiversity Biotic and Abiotic

Bryophyta

Budding Camouflage Cattle Farming Animal Husbandry Cell Structure and Function Cell Wall and Cell Membrane Cells Chloroplasts Chromatin Classification of Animal Kingdom Composition of Lymph Conduction of Nerve Impulse Cyclic and Non Cyclic Cyclic Photophosphorylation Cytoplasm Structure Function Difference Between Active and Passive Transport Difference Between Archaea And Bacteria Difference Between Blood And Lymphth Difference Between Cerebellum And Cerebrum Difference Between Chromosome And Chromatid Difference Between Diffusion and Osmosis Difference Between DNA and RNA Difference Between Haploid And Diploid Difference Between Plant Cell and Animal Cell Difference between Renewable and Non Renewable Resources Difference Between Unicellular And Multicellular Organisms Difference Between Vertebrates and Invertebrates Difference between Voluntary and Involuntary Muscles Difference Between Xylem and Phloem Different Types Ecological Pyramids Diffusion - Means of Transport Disadvantages of Dams Ecology Endemic Species Endocrine System Ductless Glands Energy Flow In Ecosystem Excretion and Its Importance Food Production

Food Web

Fragmentation

Gemmules

Grassland Adaptations

Gymnosperms

Harmful Microorganisms

Homeostasis

Human Body Anatomy

Invertebrates

Kingdom Monera Protista Fungi Law of Segregation and Law of Dominance

Life process

Light-dependent Reactions

Lipids

Living Things

Macromolecule Meiosis 1 Stages and Process Mendelian Disorders Microbodies Modern Synthetic Theory of Evolution Monera Nephron Function Renal Tubules Nuclear Membrane Nucleolus Nucleus Our Environment Parasitism Parts of Plants Passive Transport Plant Cell Plant Kingdom Plantae Plant Respiration Plants Plastids Protista Red Data Book Reproductive System of Earthworm Ribosomes Scientific Names of Animals and Plants Seed germination Sensory Perception Snail Skeletal System Spermatogenesis Structure Of Eye Taxonomic Hierarchy Taxonomy Algae Thallophyte

Types of Soil

Vacuoles What Is Adaptation Agricultural Practices Animal Kingdom Alimentary Canal Coelenterata Conservation of Forest and Wildlife Difference between Endosmosis and Exosmosis Difference between Afforestation and Deforestation Difference between Algae and Fungi Difference between Cereals and Pulses Difference Between Fragmentation and Regeneration Difference Between Neurosis and Psychosis Difference Between Nucleotide and Nucleoside

Difference Between Sea and Ocean

Difference Between Turtles and Tortoises

Difference Between Wildlife Sanctuary and National Park

Modes of Plant Reproduction

Ecological Pyramid and Its Types

Double Fertilization in Angiosperms

Ecosystem Enzymes

Fertilisation in Plants

Five Kingdoms Classification

Ganongs Potometer

Health and Hygiene

Herbivores and Carnivores

Heterotrophic Nutrition Human Digestive System

Vertebrates and Invertebrates

Morphology of Leaves

Modifications of Root

Phylum Aschelminthes

Diffusion

Osmosis

Prokaryotic Cells

Reproduction

Flora And Fauna

Overview of Food Chain

Difference Between Rabi And Kharif Crops

Vegetative Propagation Angiosperms Characteristics Soil Profile

Types of Pollution

Human Excretory System Sense Organs Rainwater Harvesting

Prokaryotic And Eukaryotic Cells

Living And Non Living Things

The Nucleus

Irrigation

Endoplasmic Reticulum

Between Data And Information

Mitochondria

Nitrogen Cycle

Diagram of Animal Cell

Meristematic Tissue

Law of Segregation Law of Dominance

Eukaryotic Cells

Sources Animal Plant Products

Photosynthesis Respiration Ozone Layer Depletion Kingdom Fungi Aerobic Anaerobic Respiration Cell Organelles Peristalsis Bacteria Oogenesis

Leaves Morphology Types Modification

Mitosis and Meiosis

How Do Organisms Reproduce

Diagram of Stomata

Monohybrid Cross Inheritance One Gene Mechanism of Breathing

Soil Erosion

Plasmolysis

Urine Formation Osmoregulation

Porifera

Flagella

Amoeba

Aerobic Respiration

Pteridophyta

Arteries And Veins Difference

Reflex Action

Stomata

Cell Wall

Diagram of Neuron

A Guide To Composition And Function Of Lymph

Difference Between Ligaments And Tendons

Insectivorous Plants

Nutrition Modes Living Organisms

Mendel Laws Of Inheritance

Nutrition In Amoeba

Calorific Value

Air Pollution Control

Fertilization In Plants

Binary Fission

Omnivores

Saprophytes

Asexual Reproduction

Deficiency Diseases

Light Reaction Vs Dark Reaction

Global Warming

Wildlife Sanctuary

Viviparous Oviparous Embryo Development

Food Preservation Methods Food Poisoning

Crops

Biofertilizers

Biodiversity Conservation

Difference Between Endocrine And Exocrine Glands

Greenhouse Effect Gases

Diagram Of Digestive System

Crop Production And Management

Types Of Pollination

Introduction and Characteristics of Living Beings Meristematic Cells Taxonomical Aids _Flora, Manual, Monograph And Catalogues Types of leaves structure of skeletal muscle spirometry human respiratory system herbarium dinoflagellates general characters and structure of leaf halophiles and their features Exchange of Gases Simple Permanent tissue Morphology of root and its types Complex permanent tissues Regions of Root History of classification and its need Breathing And Exchange Of Gases Transport of Gases Regulation of Respiration Disorders of Respiratory System Interaction of Haemoglobin with Other Gases Amniocentesis Anatomy of flowering plants ASSISTED REPRODUCTIVE TECHNOLOGIES (ARTs) Meristematic Cells Phloems Population explosion Reproductive Health RCH (Reproductive and Child Health Care) Medical termination of Pregnancy (MTP) Plant kingdom Biological Systems of Classification & Branches of Taxonomy Pteridophytes Gymnosperms Angiosperms Life cycle Patterns Anatomy of Roots Contraceptives Anatomy of stem Algae Bryophytes Tissues and its types STDs (Sexually Transmitted Diseases) Cell - The Basic Unit of Life & its Historical background Cell theory and its modification An Overview of Cell Secondary growth in dicot stem Tissue system and types Prokaryotic (Bacterial) cell Ribosomes Eukaryotic cell Cell: The basic unit of life Cytoskeleton Centrosome and Centriole Cilia and flagella Nucleus Chromosomes Evolution Movement and Locomotion Human Reproduction Absorption and Translocation of Solutes Soil as a reservoir of Essential Elements LOCOMOTION AND MOVEMENT Mechanism of Muscle Contraction Morphology in Flowering Plants Contractile proteins Muscular System Thyroid Puberty Hormones Origin of Universe Hypothalamus Cell wall Mitochondria Special Chromosomes Microbodies Introduction, Hydroponics and the study of mineral requirements of the plants Criteria of essentiality and Categorisation of Essential elements Deficiency symptoms of essential elements and toxicity The Stem The Root Cytoplasm and cell membrane Mineral Nutrition Disorders Related To Reproductive System Phylum Ctenophora Role of Macronutrients Role of Micronutrients Phylum - Coelenterata / Cnidaria Reptilia Aves Mammalia Anatomy of the leaf Animal kingdom Phylum Annelida Phylum Mollusca Phylum Echinodermata Vertebrata Chordates Disorders of circulatory system Abiotic Factors Male Reproductive System Kingdom Animalia Responses to Abiotic Factors Female Reproductive System Mammary Glands Pregnancy and Embryonic Development Cell Cancer Disorders of the Digestive System Infertility Morphology of Cockroach Blood and Plasma Alternation of Generations and Classification Predation Phaeophyceae and Rhodophyceae Phylum Coelenterata Phylum Porifera Phylum Platyhelminthes Chromosomal Disorders Competition Taxonomic Aids Alternation of generation in Pteridophytes Disorders of muscular and skeletal system General Characteristics of Living Digestive Glands Stomach Testis Buccal cavity Seed Common Diseases in Humans Permanent Tissue Permanent Tissues Xylem Pteridophytes Characteristics Parasitism and Commensalism Water Pollution and Its Control Differentiation, Dedifferentiation and Redifferentiation Organism and its Environment Carbohydrates Greenhouse Effect Gases Amino Acids Dynamic State of Body Constituents Photosynthesis Chloroplast Features of Living Beings Blood Group Lipids Kingdom Protista dense connective tissue connective tissue cartilage connective tissue blood plant cell viruses the stem Connective Tissue Bone Viroids Prions Lichens Animal cell Prokaryotic Cell Taxonomic Aids Herbarium and Museum Sphygmomanometer and ECG Population Attributes Noise pollution Diaphragm Parts of a Plant Photoperiodism Poultry Farming Nucleic Acids Cell Cycle Principles of Biotechnology Antibiotics IVF Glands Kranz Anatomy Sexual Reproduction in Plants Cytokinin Plant Development Biofertilizers Photorespiration Auxin Nutrient Cycling Types of Flowers Taxonomic Hierarchy and Categories Epidermal Tissue System Calyx and Corolla Ethylene Fertilization and Implantation Types of Fruits Inflorescence Types of Growth Macronutrients Family Liliaceae ETC and Oxidative Phosphorylation Classification of Wood Ground Tissue System Secondary Growth in Stem Plant Vascular System Electron Transport System Transpiration Difference Between Types of Cells Ovary Structure Birth Control Epithelial Tissue Endomembrane System Cell Envelopes and Cell Membrane Muscle Types Axial Skeleton Acquired Immunity Lung Volumes and Capacities Suspension and Migration Prokaryotic Cells

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