Learn definition of continuity calculus, infinite and jump discontinuity, practice one sided continuity for piecewise functions and Graph Reflection of Discontinuity.

A function 'f' is continuous at \(x=a\), if

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

\(i.e,\) \(\lim\limits_{ x\to a}=f(a)\) \((\lim\limits_{ x\to a}f(x)\,\text{will exist only if }\lim\limits_{ x\to a^+}f(x)=\lim\limits_{ x\to a^–}f(x)).\)

- \(f(a)\) is defined as an assumption in the above definition.
- If 'f' is not continuous at \(x=a\), we say it is discontinuous at \(x=a\).

Continuity at \(x=a\)

\(\Rightarrow\,f(a)=f(a^+)=f(a^-)\)

(No break at \(x=a\))

- If a function \('f'\) is discontinuous (not continuous) at \(x=a\), then there will be a break in the graph at \(x=a\).

- If a function \('f'\) is discontinuous at \(x=a\), due to the fact that limit at \(x=a\) exists but is not equal to \(f(a)\).

\(i.e,\) \(\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to a–}f(x)\neq f(a)\)

Then, it is called removable discontinuity.

- The name removable is given because it can be removed by redefining.

\(f(a)=\lim\limits_{x\to a}f(x)\)

\(\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to a–}f(x)\ne\,f(a)\)

- If we redefine \(f\) at \(x=a\) i.e \(f(a)=\lim\limits_{x\to a}f(x)\) the dot gets filled and graph becomes continuous.

**Note :** we have to evaluate the limit in such problem but the language is of continuity.

If the discontinuity at \(x=a\) is due to the fact that limit doesn't exist at \(x=a,\,i.e\)

\(\lim\limits_{x\to a^+}f(x)\ne\lim\limits_{x\to a^-}f(x)\), we say that \('f'\) has a jump discontinuity at \(x=a\).

- Such a discontinuity is reflected by a jump in the graph of function at \(x=a\). (therefore the name).
- \(f(x)=[[x]]\) has such a discontinuity at every integer.

A \(f\) has a removable discontinuity at \(x=0\)

B \(f\) has a jump discontinuity at \(x=0\)

C \(f\) is continuous at \(x=0\)

D \(f\) has an infinite discontinuity at \(x=0\)

- If the discontinuity is due to the fact that \(\lim\limits_{x\to a}f(x)=\infty\) (not a finite number) then, it is called infinite discontinuity.
- \(f(x)=tan\,x\) has an infinite discontinuity at \(x=(2n+1)\dfrac{\pi}{2}\) where, \(n\) is an integer.

\(f(x)=tan\,x\) at \(x=\dfrac{\pi}{2}\)

\(\lim\limits _{x\to{\pi}/{2}^+}tan\,x=–\infty\) and \(\lim\limits _{x\to{\pi}/{2}^–}tan\,x=\infty\)

At least one of two limits R.H.L or L.H.L is infinity.

A \(f(x)=\dfrac{1}{x+3}\) at \(x=–3\)

B \(f(x)=\dfrac{1}{x}\) at \(x=2\)

C \(f(x)=\dfrac{2}{x^2}\) at \(x=2\)

D \(f(x)=\dfrac{1}{x+7}\) at \(x=1\)

- A function \('f'\) is said to be continuous from the right or right continuous at a value \(x=a\) if

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

i.e R.H.L = \(f(a)\). (L.H.L is a different value)

- Similarly, it is said to be continuous from the left or left continuous at a value \(x=a\) if

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

i.e L.H.L = \(f(a)\), (R.H.L is a different value)

- \(f(x)=[[x]]\) is continuous from right at every integral value of \(x\).

- A function \('f'\) is said to be continuous from the right or right continuous at a value \(x=a\) if

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

i.e R.H.L = \(f(a)\). (L.H.L is a different value)

- Similarly, it is said to be continuous from the left or left continuous at a value \(x=a\) if

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

i.e L.H.L = \(f(a)\), (R.H.L is a different value)

- \(f(x)=[[x]]\) is continuous from right at every integral value of \(x\).

- If \('f'\) is a piecewise function then we can find the right and left hand limits at all point when the definition expression of f changes.
- e.g

\(f(x)= \begin{cases} 2x+5 & if & x<1\\ x^2+4 & if & x \geq1\\ \end{cases}\)

At x=1,

\(\lim\limits_{x\to 1^+}(x^2+4)=5\)= R.H.L

\(\lim\limits_{x\to 1^-}(2x+5)=7\) = L.H.L

\(f(1)=1+4=5\)

\(\therefore\,f(1)=\) R.H.L \(\ne\) L.H.L

\(\therefore\,f\) is right continuous at \(x=1\).

A \(f\) is right continuity at \(x=1\)

B \(f\) is left continuity at \(x=1\)

C \(f\) is continuous at \(x=1\)

D Nothing can be said about continuity of \('f'\) at \(x=1\)

Consider the graphs of functions having various types of discontinuities.

\('f'\) has a removable discontinuity at \(x=a\) in the graph shown.

\('f'\) has infinite discontinuity at \(x=a\) in the graph shown.

R.H.L \(\ne\) L.H.L (R.H.L – L.H.L) = Jump.

\(f\) has a jump discontinuity at \(x=a\), there is a jump in the graph at \(x=a\).