Continuity

Continuity

1. Definition — Joint Continuity

$f(x,y)$ is continuous at $(a,b)$ if:

i.e. the joint limit exists and equals the actual value. In ε-δ form, near $(a,b)$ the values stay within $ε$ of $f(a,b)$. If $f$ is continuous at every point of a domain $D$, it is continuous on $D$.

If any of: limit does not exist, limit exists but ≠ $f(a,b)$, or $f(a,b)$ undefined — then discontinuous at $(a,b)$. For points where $f$ is defined by a formula whose denominator ≠0, continuity is automatic (polynomials, $e^{x+y}$, $sin x$ etc. are continuous everywhere).

2. Continuity via Paths and Polar — How to Test

Same tools as limits: two-path difference ⇒ discontinuity; polar/squeeze showing limit equals value ⇒ continuity. Removable discontinuity: define $f(a,b)$ to fill the limit and it becomes continuous.

3. Worked Examples

(a) f(x,y)= x y/(x²+y²) for (x,y)≠(0,0), f(0,0)=0:
    Limit at (0,0) does not exist (path y=mx gives m/(1+m²)) ⇒ discontinuous at (0,0) ✓
    No way to define value to make it continuous.

(b) f(x,y)= x² y/(x²+y²) for (x,y)≠(0,0), f(0,0)=0:
    |f|≤|y|→0 ⇒ limit 0 = f(0,0) ⇒ CONTINUOUS at (0,0) ✓
    In polar: r cos² sin →0 uniformly.

(c) f(x,y)= (x²+y²) sin(1/(x²+y²)), f(0,0)=0:
    |f| ≤ x²+y² = r² →0 ⇒ continuous at 0 (squeeze: bounded sin factor) ✓

(d) f(x,y)= (x³ - y³)/(x²+y²) , f(0,0)=0: polar → r(cos³ - sin³)→0 ⇒ continuous ✓

(e) f(x,y)= 1/(x²+y²) at (0,0) undefined and limit ∞ ⇒ discontinuous (infinite) ✓

(f) Piecewise: f= (x y²)/(x²+y⁴) , f(0,0)=0 : along y²=mx? Path y²=mx gives m/(1+m²)?? Actually along x=y²: f= y⁴/(y⁴+y⁴)=1/2 ≠0 ⇒ discontinuous ⇒ shows parabola path needed beyond lines ✓

4. Properties — What Continuity Gives

• Sums/products/quotients (denominator ≠0) of continuous functions are continuous.
• Composition: if $g$ continuous at $(a,b)$ and $h$ continuous at $g(a,b)$, then $h∘g$ continuous — e.g., $e^{x y}$ continuous everywhere.
• On a closed bounded domain, continuous $f$ attains max/min and is uniformly continuous — used in maxima theory later.

5. Relation to Partial Derivatives (preview)

Continuity does not imply existence of partial derivatives, and existence of partials does not imply continuity — e.g., $f=xy/(x²+y²)$ with $f(0,0)=0$ has $f_x(0,0)=0, f_y(0,0)=0$ but is discontinuous. The stronger condition that guarantees continuity is differentiability (next topic), which needs continuous partials.

6. Exam Points

• Test continuity at $(0,0)$ for $xy/(x²+y²)$, $x²y/(x²+y²)$, $x²y²/(x⁴+y²)$ type — 5-7 marks, very frequent.
• Show two-path failure ⇒ discontinuity — 3-5 marks.
• Prove continuity via squeeze/polar and compute limit value — 5 marks, frequent.
• State definition and give example where partials exist but $f$ discontinuous — 5 marks, frequent.