Let \('f'\) and \('g'\) be two functions which are continuous at \(x=a\), then the following functions will also be continuous at \(x=a\).

1. \(f+g\)
2. \(f-g\)
3. \(cf\)
4. \(fg\)
5. \(\dfrac{f}{g}\) if  \(g(a)\ne0\)

• We say that :

1. Sum of two continuous function is also continuous.

Proof: Let \(h(x)=f(x)+g(x)\,\forall \,x\)

and at \(x=a\)  \(f,\,g\) are say both continuous.

Then,  \(\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to a^-}f(x)=f(a)\)

and  \(\lim\limits_{x\to a^+}g(x)=\lim\limits_{x\to a^-}g(x)=g(a)\)

\(h(a)=f(a)+g(a)\)

\(\lim\limits_{x\to a^+}h(x)=\lim\limits_{x\to a^+}(f(x)+g(x))=\lim\limits_{x\to a^+}f(x)+\lim\limits_{x\to a^+}g(x)\)

\(\lim\limits_{x\to a^-}h(x)=\lim\limits_{x\to a^-}(f(x)+g(x))=\lim\limits_{x\to a^-}f(x)+\lim\limits_{x\to a^-}g(x)\)

\(\therefore\,\,\lim\limits_{x\to a^+}h(x)=\lim\limits_{x\to a^-}h(x)=h(a)\)

\(\therefore\,\,h\) is also continuous at \(x=a\)

Similarly,

2.Difference of two continuous functions is continuous.

3.Product of two continuous functions is continuous.

4.Multiplication by a constant will not change the continuous behavior of a function.

5.Ratio of two continuous functions is continuous. (except where denominator is 0).

1. All polynomials are continuous every where i.e in \((–\infty,\,\infty)\).
2. All rational functions are continuous whenever they are defined i.e they are continuous in their domain.

• e.g \(f(x)=2x^3+x^2+x-1\) is continuous everywhere

\(f(x)=\dfrac{2x+1}{x-2}\) is continuous for all \(x\) except \(\underbrace{x=2}_{not \,in \,the\,domain}\) 

If  \('g'\) is continuous at \(x=a\) and \('f'\) is continuous at \(x=g(a)\), then the composite function \(f(g(x))\) is continuous at \(x=a\).

• e.g  \(h(x)=cos\,x^3\) is continuous at all values of \(x\), because \(h(x)=f(g(x))\) where, \(g(x)=x^3\) and \(f(x)=cos\,x\), both of which are continuous for all \(x\).
• Composite function rule extends for composite of more than two function. 

\(\lim\limits_{x\to a}f(g(x)) =f\left(\lim\limits_{x\to a }g(x)\right)\)= \(f(b)\). If

(1)  \(\lim\limits_{x\to a}\,g(x)=b\) 

  (2)  \('f'\)is continuous at \(x=b\). 

e.g 

Consider,  \(\lim\limits_{x\to a}\,cos(x-2)\)

which is of the form  \(\lim\limits_{x\to a}\,f(g(x))\)

where, \(g(x)=x-2\) and \(f(x)=cos\,x\)

\(\therefore\,\lim\limits_{x\to 2}\,cos(x-2)\)

\(=cos\left(\lim\limits_{x\to 2}\,(x-2)\right)\)

\(=cos\,0=1\)

All trigonometric functions are continuous at all points where they are defined i.e they are continuous in their domain.

• \(sin\,x,\,cos\,x\) are continuous in R
• \(tan\,x,\,sec\,x\) are continuous in \(R-\left\{(2n+1)\dfrac{\pi}{2}\right\}\)\((n\in I)\)
• \(cosec\,x,\,cot\,x\) are continuous in \(R-\{n\pi\}\) \((n\in I)\)

The root function \(f(x)= \sqrt[n]{x}=x^{1/n}\) is continuous for all values of  \(x\) where it is defined or continuous in its domain.

• Consider the graph of \(f(x)=x^{1/n}\).
• For the case, when \(n\) is even, the graph is similar to  \(f(x)=\sqrt x\) as shown in figure.

• For the case, when \(n \) is odd the graph is the graph is similar to \(f(x)=\sqrt[3]{x}\)  as shown in the graph.

• In both cases \(f(x)=x^{{1}/{n}}\) is continuous in domain.

A function \('f'\) is said to be continuous in an interval

[a, b] if it is continuous for all values of  \(x\) in [a, b].

• At \(x=a\), continuity means \(\lim\limits_{x\to a^+}\,f(x)=f(a)\) (continuity from right).
• At \(x=b\), continuity means \(\lim\limits_{x\to b^-}\,f(x)=f(b)\) (continuity from left).
• At all the interior points the definition of continuity remain the same i.e R.H.L=L.H.L = \(f(\alpha)\)

Where \(x=\alpha\) is any interior point.

• Even a single point of discontinuity in any interval will make the function discontinuous in the entire interval.

Suppose \('f'\) is a piecewise defined function.

\(f(x)= \begin{cases} 2x+1 & if & x<2\\ x + 3 & if & x\geq0 \end{cases}\)

then to test continuity at \(f\) at \(x=2\)

R.H.L = \(\lim\limits_{x\to 2^+}f(x)=\lim\limits_{x\to 2^+}(x+3)=5\;\;\;\;\;\;(2^+>2)\)

L.H.L =  \(\lim\limits_{x\to 2^-}f(x)=\lim\limits_{x\to 2^-}(2x+1)=5\,\;\;\;\;(2^-<2)\)

\(f(2)=2+3=5\)

\(\therefore\) \(f\) is continuous at \(x=2\) 

Now sometimes it is asked after  giving continuity ,what is the value of certain parameter.

Consider

\(f(x)= \begin{cases} 2x+\alpha & if & x<2\\ x + 3 & if & x\geq2 \end{cases}\)  is continuous at \(x=2\) 

then R.H.L = L.H.L = \(f(2)\Rightarrow\,\alpha=1\)