if a point is not continuous can it be differentiable?
> No, a point on a function cannot be differentiable if it is not continuous at that point. Differentiability implies continuity, so if a function is discontinuous, it is also not differentiable. A function can only be differentiable if the limit of the function exists at that point and is equal to the function's value, which is the definition of continuity.
📘 Note: Making a Function Continuous
1. What is a Piecewise Function?
👉 A function defined in different ways for different intervals.
Example: f(x) = { x^2 if x<1 kx+1 if x≥1 }
Here the rule changes at x=1. (1 is the Breakpoint)
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2. Condition for Continuity at a Point
A function f(x) is continuous at x=a if:
lim (x→a^−) f(x) = lim (x→a^+) f(x) = f(a)
- Left-hand limit = value from the left side of a.
- Right-hand limit = value from the right side of a.
- f(a) = actual defined value at a.
👉 If all three are equal, the function is continuous at x=a.
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3. When Unknown Constants Are Involved
👉 In piecewise functions, sometimes a constant is unknown (like k). We solve for it by equating the left-hand and right-hand limits.
Example: f(x) = { x^2, if x<2 kx+1, if x≥2 }
Step 1: Find left limit (x→2^−): lim (x→2^−) f(x) = 2^2 = 4
Step 2: Find right limit (x→2^+): lim (x→2^+) f(x) = k(2)+1 = 2k+1
Step 3: Set equal (for continuity): 4 = 2k+1 ⇒ k=3/2
Step 4: Define f(2): f(2) = 2(3/2)+1 = 4 (matches the limit). ✅ Continuous
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4. Where Is This Applied?
- Mathematics: In calculus, to ensure limits and derivatives exist.
- Engineering/Physics: To model systems without sudden jumps (temperature change, force, voltage).
- Computer Science: In graphing, interpolation, or ensuring smooth transitions.
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5. Your Question 🤔
> "If making a function continuous means adding another rule where x does not blow up, isn’t that the same as it being not continuous? Aren’t we just putting a point on another point where it doesn’t blow up?"
👉 Answer:
- Yes, you’re right that if a function “blows up” (→∞) at a point, it can never be made continuous by just defining f(a). Example: f(x)=1/x at x=0. No finite f(0) can fix it.
- But if the function has a removable hole (limit exists, but f(a) missing or wrong), then we can “patch” it by defining f(a)=that limit. Example: f(x)=(x^2-1)/(x-1) has a hole at x=1, but if we define f(1)=2, it becomes continuous.
⚡ So:
- Jump/blow up → not fixable
- Hole/removable discontinuity → fixable by defining f(a)=limit
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✅ In short: “Making a function continuous” means patching removable holes, not fixing vertical asymptotes.
📘 Why Assigning f(a)=limit Works
1. What Continuity Really Means
f(x) is continuous at x=a if:
lim (x→a) f(x) = f(a)
- lim (x→a) f(x) exists → the function “wants” to approach some number L.
- f(a) must equal that L.
If f(a) ≠ L or f(a) is undefined, continuity fails.
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2. Removable Discontinuity
Example: f(x) = (x^2-1)/(x-1)
- Factor: f(x) = (x-1)(x+1)/(x-1) = x+1, for x≠1
- So, lim (x→1) f(x) = 1+1 = 2 [ Think again Here, This is Not Magic that I pull '2' out of nowhere its x + 1 ]
✅ The limit exists. The function has a hole at x=1.
- Original f(1) is undefined → discontinuous
- If we define f(1) = 2, now:
lim (x→1) f(x) = f(1) = 2
Boom ✅, continuous.
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3. Why This Works (“Proof”)
- The limit exists → the graph is approaching a number L.
- Assigning f(a)=L fills the hole exactly where the graph wants to go.
- Formally, by the definition of continuity:
lim (x→a) f(x) = L and f(a) = L ⇒ continuity satisfied
So we’re not arbitrarily picking a number. We are using the number the function is already approaching. That’s the proof.
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4. When You Cannot Fix Continuity
- If lim (x→a) f(x) does not exist, no choice of f(a) can make it continuous.
- Example: f(x) = 1/x at x=0 → lim (x→0^−) = -∞, lim (x→0^+) = +∞ → discontinuity cannot be “fixed”
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✅ Key Idea:
> “Making a function continuous” = patching a removable discontinuity, not altering the limit itself. The limit gives the exact value we must assign. ----------- # Breakpoint details ----------- # 📘 Making a Function Continuous (Breakpoint / Stepwise Notes)
1. Definition
A function f(x) is continuous at x = a if:
lim (x→a⁻) f(x) = lim (x→a⁺) f(x) = f(a)
- x→a⁻ → left-hand limit (approach a from left)
- x→a⁺ → right-hand limit (approach a from right)
- f(a) → value defined at a
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2. Breakpoints / Removable Discontinuities
- A breakpoint is a point x = a where the function is not continuous.
- Types of breakpoints:
1. Removable discontinuity (hole): lim (x→a) exists, f(a) undefined or ≠ limit → can “patch” 2. Jump discontinuity: lim (x→a⁻) ≠ lim (x→a⁺) → cannot patch 3. Infinite / vertical asymptote: lim → ∞ or -∞ → cannot patch
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3. Stepwise Method to Make Function Continuous
Example: f(x) = { x² , x<2 ; kx+1 , x≥2 }
Step 1: Find Left-Hand Limit (LHL) lim (x→2⁻) f(x) = 2² = 4
Step 2: Find Right-Hand Limit (RHL) lim (x→2⁺) f(x) = k×2 + 1 = 2k+1
Step 3: Equate for Continuity LHL = RHL ⇒ 4 = 2k+1 ⇒ k = 3/2
Step 4: Assign f(a) = limit f(2) = lim (x→2) f(x) = 4
✅ Function is now continuous at x=2.
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4. Important Notes About Breakpoints
- We are not arbitrarily picking f(a).
- Proof: continuity definition demands f(a) = lim (x→a) f(x).
- Only removable holes can be patched.
- Jump or infinite discontinuities cannot be made continuous by assigning a value.
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5. Shortcut / Quick Recall
- Check if limit exists at breakpoint.
- If yes → find value → assign f(a)=limit
- If no → cannot patch → discontinuity remains
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