\displaystyle 1.\ \text{A real valued function }f(x)\text{ is continuous at a point }a\text{ in its domain iff }
\displaystyle \lim_{x\to a}f(x)=f(a).
\displaystyle \text{i.e. the limit of the function at }x=a\text{ is equal to the value of the function at }x=a.

\displaystyle 2.\ \text{A function }f(x)\text{ is said to be continuous if it is continuous at every point of its domain.}

\displaystyle 3.\ \text{Sum, difference, product and quotient of continuous functions are continuous i.e. if }f(x)
\displaystyle \text{and }g(x)\text{ are continuous functions on their common domain, then }f\pm g,\ fg,\ \frac{f}{g},\ kf
\displaystyle \text{(}k\text{ is a constant) are continuous.}

\displaystyle 4.\ \text{Let }f\text{ and }g\text{ be real functions such that }fog\text{ is defined. If }g\text{ is continuous at }x=a \text{and }f\text{ is continuous at }g(a),\text{ then }fog\text{ is continuous at }x=a.

\displaystyle 5.\ \text{Following functions are everywhere continuous:}
\displaystyle (i)\ \text{A constant function}\qquad (ii)\ \text{The identity function}
\displaystyle (iii)\ \text{A polynomial function}\qquad (iv)\ \text{Modulus function}
\displaystyle (v)\ \text{Exponential function}\qquad (vi)\ \text{Sine and Cosine functions}

\displaystyle 6.\ \text{Following functions are continuous in their domains:}
\displaystyle (i)\ \text{A logarithmic function}
\displaystyle (ii)\ \text{A rational function}
\displaystyle (iii)\ \text{Tangent, cotangent, secant and cosecant functions}

\displaystyle 7.\ \text{If }f\text{ is continuous function, then }|f|\text{ and }\frac{1}{f}\text{ are continuous in their domains.}

\displaystyle 8.\ \sin^{-1}x,\ \cos^{-1}x,\ \tan^{-1}x,\ \cot^{-1}x,\ \mathrm{cosec}^{-1}x\text{ and }\sec^{-1}x \text{are continuous functions} \\ \text{on their respective domains.}

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