## Continuity and Differentiability

# Continuity

- A real valued function is
at a point in its domain if the limit of the function at that point equals the value of the function at that point. A function is continuous if it is continuous on the whole of its domain.**continuous** - A function f(x) is said to be continuous at a point x = a, if \lim_{x \rightarrow a}=f(a)
- A function f(x) is said to be continuous at a point x = a, if LHL = RHL = f(a)

\Rightarrow \lim_{x \rightarrow a^{-}}f(x)=\lim_{x \rightarrow a^{+}}f(x)=f(a)

\Rightarrow \lim_{h \rightarrow 0}f(a-h)=\lim_{h \rightarrow 0}f(a+h)=f(a) - A function f(x) is said to be left continuous at x = a, if \Rightarrow \lim_{x \rightarrow a^{-}}f(x)=f(a) i.e., LHL = f(a)
- A function f(x) is said to be right continuous at x = a, if \Rightarrow \lim_{x \rightarrow a^{+}}f(x)=f(a) i.e., RHL = f(a)
- A function f(x) is said to be continuous on (a, b) if f(x) is continuous at every point in (a, b). i.e., ∃ an arbitrary point c ∈ (a, b), f(x) is continuous at x = c.
- f(x) is continuous on [a, b] if

(i) fx) is Right continuous at x = a. i.e., \lim_{x \rightarrow a^{+}}f(x)=f(a)

(ii) f(x) is left continuous at x = b i.e., \lim_{x \rightarrow b^{-}}f(x)=f(b)

(iii) f(x) is continuous on (a, b) - The graph of the curve is said be continuous function through out the domain, the graph has no holes (or) no gaps.
**Properties of continuous functions:**

→ If f and g are two continuous functions, then f + g, f − g, fg, \frac{f}{g} are also continuous.

→ The composite function of two continuous functions is also continuous.

**Tips for continuity of functions:**

- A function f(x) is continuous on R, then |f(x)| is also continuous on R but converse need not be true.
- Every polynomial function is continuous on R.
- Every constant function is continuous on R.
- A function f(x) = [x] (integral part of x) is continuous all real numbers except integers.
- Every exponential function and logarithmic function is continuous on its domains.
- → sin x is continuous on R

→ cos x is continuous on R - → tan x is continuous on R-\left\{(2n+1)\frac{\pi}{2}\right\}, n \ \epsilon \ z

→ sec x is continuous on R-\left\{(2n+1)\frac{\pi}{2}\right\}, n \ \epsilon \ z - → cot x is continuous on R − {nπ}, n ∈ z

→ cosec x is continuous on R − {nπ}, n ∈ z - A function f(x) is not continuous, then it is said to be discontinuous
**Missing point discontinuity:**\lim_{x \rightarrow a} \ f(x) exists but f(a) does not exists.

\therefore \lim_{x \rightarrow a} \ f(x) \neq f(a)**Isolated point discontinuity:**\lim_{x \rightarrow a} \ f(x) exists but it is not equal to f(a) even f(a) exists also

\therefore \lim_{x \rightarrow a} \ f(x) \neq f(a)**Finite type**: If both LHL and RHL exists finitely but both are not equal i.e., LHL ≠ RHL**Infinite type:**If atleast one of the limits LHL (or) RHL goes to ∞ i.e., either LHL = ∞ (or) RHL = ∞.**Oscillatory type:**Limit value is oscillating between two finite values but not getting a particular constant.**Jump discontinuity:**The non negative difference of both LHL and RHL is called Jump discontinuity i.e.,

Jump = |LHL − RHL|

If both LHL and RHL must be finite.

### View the Topic in this video From 00:40 To 25:55

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- Sum, difference, product and quotient of continuous functions are continuous. i.e., if
*f*and*g*are continuous functions, then

(*f ± g*) (*x*) =*f*(*x*) ±*g*(*x*) is continuous.

(*f . g*) (*x*) =*f*(*x*) .*g*(*x*) is continuous.

\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)} (wherever g(x) ≠ 0) is continuous.