Continuous and Discontinuous Functions

A real-valued function \[f\] is said to be continuous at \[x = c\] if

\[ \boxed{\lim_{x \to c} f(x) = f(c)} \]

In terms of one-sided limits,

\[ \boxed{\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)} \]

Thus, for continuity at \[x = c\]:

• \[f(c)\] must be defined.
• Left-hand limit must exist.
• Right-hand limit must exist.
• Both limits must be equal to \[f(c)\]

If any of these conditions fails, \[f\] is discontinuous at \[x = c\].

A real function \[f\] is said to be a continuous function if it is continuous at every point in its domain.

\[ \boxed{\lim_{x \to c} f(x) = f(c)} \] for every \[c\] in the domain of \[f\].

Continuity at End Points

If \[f\] is defined on a closed interval \[[a, b]\]:

At the left endpoint \[a\], \[ \boxed{\lim_{x \to a^+} f(x) = f(a)} \]

At the right endpoint \[b\], \[ \boxed{\lim_{x \to b^-} f(x) = f(b)} \]

Only the appropriate one-sided limit is considered at an endpoint.

Removable Discontinuity:

If \[\lim_{x\to a^{-}}f\left(x\right)=\lim_{x\to a^{+}}f\left(x\right)\neq f\left(a\right),\] then f(x) is said to be removable discontinuous.

Non Removable Discontinuity:

If \[\lim_{x\to a^{+}}f\left(x\right)\neq\lim_{x\to a^{-}}f\left(x\right),\] then f(x) is said to be non-removable discontinuous.

At \[x = c\]:

• Step 1: Find \[f(c)\].
• Step 2: Find \[ \lim_{x \to c^-} f(x). \]
• Step 3: Find \[ \lim_{x \to c^+} f(x). \]
• Step 4: Compare them.

If \[ \boxed{\text{LHL} = \text{RHL} = f(c)} \] then \[f\] is continuous at \[x = c\].

Otherwise, \[f\] is discontinuous there.

A function is continuous at a point if its graph can be drawn near that point without lifting the pen from the paper.

For example,

\[ f(x) = \begin{cases} 1, & x \leq 0 \\ 2, & x > 0 \end{cases} \]

is discontinuous at \[x = 0\], because

\[ \lim_{x \to 0^-} f(x) = 1 \] and \[ \lim_{x \to 0^+} f(x) = 2. \]

Since LHL \[\neq\] RHL, the limit does not exist.

Constant Function: 

\[ f(x) = k \] is continuous for every real number.

Identity Function:

\[ f(x) = x \] is continuous for every real number.

Polynomial Function:

Every polynomial function is continuous for all real numbers.

For example, \[ f(x) = x^3 + x^2 - 1 \] is continuous for every \[x \in \mathbb{R}\].

Reciprocal Function:

\[ f(x) = \frac{1}{x} \] is continuous at every point in its domain, i.e. \[ x \neq 0. \] 

It is not defined at \[x = 0\].

Piecewise Function:

A function defined by different formulas for different parts of its domain. Continuity is checked especially at the points where the formula changes.

Greatest Integer Function:

Examine whether the function \[f(x) = x^2\] is continuous at x = 0.

• Value at point: \[f(0) = 0^2 = 0\]

• Limit at point: \[\lim_{x \to 0} x^2 = 0^2 = 0\]

• Since \[\lim_{x \to 0} f(x) = f(0)\], the function is continuous at x = 0.

Is the function defined by f(x) = | x |, a continuous function?

Solution:

For \[ f(x) = |x| = \begin{cases} -x, & x < 0 \\ x, & x \geq 0 \end{cases}, \]

at \[x = 0\],

\[ f(0) = 0 \]

\[ \lim_{x \to 0^-} f(x) = 0 \] and \[ \lim_{x \to 0^+} f(x) = 0. \]

Therefore, \[ \text{LHL} = \text{RHL} = f(0) = 0 \]

Hence, \[f(x) = |x|\] is continuous at \[x = 0\].

Show that the function \[f\] given by

\[ f(x) = \begin{cases} x^3 + 3, & \text{if } x \neq 0 \\ 1, & \text{if } x = 0 \end{cases} \]

is not continuous at \[x = 0\].

Solution:

Consider \[ f(x) = \begin{cases} x^3 + 3, & x \neq 0 \\ 1, & x = 0 \end{cases}. \]

At \[x = 0\], \[ f(0) = 1 \]

but \[ \lim_{x \to 0} f(x) = \lim_{x \to 0}(x^3 + 3) = 3. \]

Thus, \[ \lim_{x \to 0} f(x) \neq f(0). \]

Therefore, \[ f\ \text{is discontinuous at}\ x = 0 \]

•  Continuity at \[x = c\]: \[ \boxed{\lim_{x \to c} f(x) = f(c)} \]
• Practical test: \[ \boxed{\text{LHL} = \text{RHL} = f(c)} \]
  If this condition fails, the function is discontinuous at \[c\].
• A function is continuous if it is continuous at every point in its domain.
• Constant, identity and polynomial functions are continuous on their domains.
• \[\dfrac{1}{x}\] is continuous for \[x \neq 0\].
• For a piecewise function, check continuity particularly at the point where the rule changes.
• The greatest integer function \[[x]\] is discontinuous at every integer.