Math Lecture 5: Continuity of Functions PDF

Lecture 5 Continuity of Functions Mohamed Sief [email protected] Nile Valley University October 31, 2024 Mohamed Sief (NVU) Mathematics October 31, 2024 1 / 23 Lecture Outline 1 Con...

Lecture 5 Continuity of Functions Mohamed Sief [email protected] Nile Valley University October 31, 2024 Mohamed Sief (NVU) Mathematics October 31, 2024 1 / 23 Lecture Outline 1 Continuity and Limits 2 Tutorials 3 Conclusion Mohamed Sief (NVU) Mathematics October 31, 2024 2 / 23 Continuity A function f (x) is continuous at a point a if: 1 f (a) is defined 2 limx→a f (x) exists 3 limx→a f (x) = f (a) If any of these conditions fail, the function has a discontinuity at a. Mohamed Sief (NVU) Mathematics October 31, 2024 3 / 23 Example 3: Identifying Discontinuities Example Where are each of the following functions discontinuous? x2 −x−2 (a) f (x) = x−2 ( 1 x2 if x ̸= 0 (b) f (x) = 1 if x = 0 ( 2 x −x−2 x−2 if x ̸= 2 (c) f (x) = 1 if x = 2 ( 0 if t < 0 (d) H(t) = 1 if t ≥ 0 Mohamed Sief (NVU) Mathematics October 31, 2024 4 / 23 Solution (a) x2 −x−2 For f (x) = x−2 : First, note that f is not defined at x = 2 The domain is all real numbers except 2 Therefore, f is discontinuous at x=2 Mohamed Sief (NVU) Mathematics October 31, 2024 5 / 23 Solution (b) ( 1 x2 if x ̸= 0 For f (x) = 1 if x = 0 f (0) = 1 is defined 1 However, let’s check limx→0 x2 1 limx→0+ x2 =∞ 1 limx→0− x2 =∞ Therefore, limx→0 f (x) does not exist f is discontinuous at x = 0 Mohamed Sief (NVU) Mathematics October 31, 2024 6 / 23 Solution (c) ( x2 −x−2 x−2 if x ̸= 2 For f (x) = 1 if x = 2 f (2) = 1 is defined For x ̸= 2, we can factor: x2 − x − 2 lim x→2 x−2 (x + 1)(x − 2) = lim x→2 x−2 = lim (x + 1) = 3 x→2 Since limx→2 f (x) = 3 ̸= f (2) = 1 f is discontinuous at x = 2 Mohamed Sief (NVU) Mathematics October 31, 2024 7 / 23 Solution (d): The Heaviside Function ( 0 if t < 0 For H(t) = 1 if t ≥ 0 Named after Oliver Heaviside (1850-1925) Used to model electric current being turned on At t = 0: limt→0− H(t) = 0 limt→0+ H(t) = 1 Since left and right limits are different H is discontinuous at t = 0 Mohamed Sief (NVU) Mathematics October 31, 2024 8 / 23 Visual Summary of Discontinuities Removable Discontinuity Limit exists at the point Function value undefined or differs from limit (f (a) undefined or f (a) ̸= lim f (x)) x→a Can be ”fixed” by redefining at a single point Mohamed Sief (NVU) Mathematics October 31, 2024 9 / 23 Visual Summary of Discontinuities Removable Discontinuity Limit exists at the point Function value undefined or differs from limit (f (a) undefined or f (a) ̸= lim f (x)) x→a Can be ”fixed” by redefining at a single point Mohamed Sief (NVU) Mathematics October 31, 2024 9 / 23 Visual Summary of Discontinuities Removable Discontinuity Limit exists at the point Function value undefined or differs from limit (f (a) undefined or f (a) ̸= lim f (x)) x→a Can be ”fixed” by redefining at a single point Infinite Discontinuity Limit is infinite ( lim f (x) = ±∞) x→a Vertical asymptote at x = a Mohamed Sief (NVU) Mathematics October 31, 2024 9 / 23 Visual Summary of Discontinuities Removable Discontinuity Limit exists at the point Function value undefined or differs from limit (f (a) undefined or f (a) ̸= lim f (x)) x→a Can be ”fixed” by redefining at a single point Infinite Discontinuity Limit is infinite ( lim f (x) = ±∞) x→a Vertical asymptote at x = a Jump Discontinuity Left and right limits exist but are different ( lim f (x) ̸= lim f (x)) x→a− x→a+ Cannot be ”fixed” Mohamed Sief (NVU) Mathematics October 31, 2024 9 / 23 Which Functions Are Continuous? Theorem (1) The following types of functions are continuous at every number in their domains: Algebraic Functions Transcendental Functions Mohamed Sief (NVU) Mathematics October 31, 2024 10 / 23 Which Functions Are Continuous? Theorem (1) The following types of functions are continuous at every number in their domains: Algebraic Functions Transcendental Functions Polynomials f (x) = axn + bxn−1 + · · · + k Mohamed Sief (NVU) Mathematics October 31, 2024 10 / 23 Which Functions Are Continuous? Theorem (1) The following types of functions are continuous at every number in their domains: Algebraic Functions Transcendental Functions Polynomials Power & Root Functions √ f (x) = xn , n x Mohamed Sief (NVU) Mathematics October 31, 2024 10 / 23 Which Functions Are Continuous? Theorem (1) The following types of functions are continuous at every number in their domains: Algebraic Functions Transcendental Functions Polynomials Power & Root Functions Rational Functions P (x) f (x) = Q(x) Mohamed Sief (NVU) Mathematics October 31, 2024 10 / 23 Which Functions Are Continuous? Theorem (1) The following types of functions are continuous at every number in their domains: Algebraic Functions Transcendental Functions Polynomials Trigonometric Functions Power & Root Functions sin x, cos x, tan x Rational Functions Mohamed Sief (NVU) Mathematics October 31, 2024 10 / 23 Which Functions Are Continuous? Theorem (1) The following types of functions are continuous at every number in their domains: Algebraic Functions Transcendental Functions Polynomials Trigonometric Functions Power & Root Functions Exponential Functions Rational Functions f (x) = ax , ex Mohamed Sief (NVU) Mathematics October 31, 2024 10 / 23 Which Functions Are Continuous? Theorem (1) The following types of functions are continuous at every number in their domains: Algebraic Functions Transcendental Functions Polynomials Trigonometric Functions Power & Root Functions Exponential Functions Rational Functions Logarithmic Functions loga x, ln x Mohamed Sief (NVU) Mathematics October 31, 2024 10 / 23 Continuity of Function Operations Theorem If f and g are continuous at a and c is a constant, then the following functions are also continuous at a: 1 f ±g (Sum &Difference Rule) Adding and Subtracting 2 cf (Constant Multiple Rule) continuous functions pre- 3 fg (Product Rule) serves continuity f Multiplying by a con- 4 if g(a) ̸= 0 (Quotient Rule) g stant preserves conti- nuity Division preserves continuity when de- Multiplying contin- nominator is non- uous functions pre- zero serves continuity Mohamed Sief (NVU) Mathematics October 31, 2024 11 / 23 Example: Continuity of Piecewise Function Function f (x) y    x2 − 1, −1 ≤ x < 0 2    2x, 0